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THEORY  AND  CALCULATION 


ALTERNATING   CURRENT 
PHENOMENA/ 


BY 

CHARLES    PROTEUS.STEINMETZ 


WITH   THE   ASSISTANCE   OF 


ERNST  J.    BERG 


THIRD  EDITION,  REVISED  AND  ENLARGED 


NEW     YORK 
ELECTRICAL   WORLD    AND   ENGINEER 

INCORPORATED 
I9OO 


COPYRIGHT,  1900, 


ELECTRICAL  WORLD  AND  ENGINEER. 
(INCORPORATED.) 


TYPOGRAPHY    BY   C.   J.    PETERS   *   SON,    BOSTON. 


1C  27 


DEDICATED 

TO   THE 

MEMORY    OF    MY    FATHER, 
CARL    HEINRICH    STEINMETZ. 


PREFACE   TO    THE  THIRD    EDITION. 


IN  preparing  the  third  edition,  great  improvements  have 
been  made,  and  a  considerable  part  of  the  work  entirely  re- 
written, with  the  addition  of  much  new  material.  A  number 
of  new  chapters  have  been  added,  as  those  on  vector  rep- 
resentation of  double  frequency  quantities  as  power  and 
torque,  and  on  symbolic  representation  of  general  alternating 
waves.  Many  chapters  have  been  more  or  less  completely 
rewritten  and  enlarged,  as  those  on  the  topographical 
method,  on  distributed  capacity  and  inductance,  on  fre- 
quency converters  and  induction  machines,  etc.,  and  the 
size  of  the  -volume  thereby  greatly  increased. 

The  denotations  have  been  carried  through  systematically, 
by  distinguishing  between  complex  vectors  and  absolute 
values  throughout  the  text ;  and  the  typographical  errors 
which  had  passed  into  the  first  and  second  editions,  have 
been  eliminated  with  the  utmost  care. 

To  those  gentlemen  who  so  materially  assisted  me  by 
drawing  my  attention  to  errors  in  the  previous  editions,  I 
herewith  extend  my  best  thanks,  and  shall  be  obliged  for 
any  further  assistance  in  this  direction.  Great  credit  is 
due  to  the  publishers,  who  have  gone  to  very  considerable 
expense  in  bringing  out  the  third  edition  in  its  present  form, 
and  carrying  out  all  my  requests  regarding  changes  and 
additions.  Many  thanks  are  due  to  Mr.  Townsend  Wolcott 
for  his  valuable  and  able  assistance  in  preparing  and  editing 
the  third  edition. 

CHARLES  PROTEUS  STEINMETZ. 

CAMP  MOHAWK,  VIELE'S  CREEK, 
July,  jgoo. 


PREFACE    TO    FIRST    EDITION. 


THE  following  volume  is  intended  as  an  exposition  of 
the  methods  which  I  have  found  useful  in  the  theoretical 
investigation  and  calculation  of  the  manifold  phenomena 
taking  place  in  alternating-current  circuits,  and  of  their 
application  to  alternating-current  apparatus. 

While  the  book  is  not  intended  as  first  instruction  for 
a  beginner,  but  presupposes  some  knowledge  of  electrical 
engineering,  I  have  endeavored  to  make  it  as  elementary 
as  possible,  and  have  therefore  only  used  common  algebra 
and  trigonometry,  practically  excluding  calculus,  except  in 
§§  106  to  115  and  Appendix  II. ;  and  even  §§  106  to  115 
have  been  paralleled  by  the  elementary  approximation  of 
the  same  phenomenon  in  §§  102  to  105. 

All  the  methods  used  in  the  book  have  been  introduced 
and  explicitly  discussed,  with  instances  of  their  application, 
the  first  part  of  the  book  being  devoted  to  this.  In  the  in- 
vestigation of  alternating-current  phenomena  and  apparatus, 
one  method  only  has  usually  been  employed,  though  the 
other  available  methods  are  sufficiently  explained  to  show 
their  application. 

A  considerable  part  of  the  book  is  necessarily  devoted 
to  the  application  of  complex  imaginary  quantities,  as  the 
method  which  I  found  most  useful  in  dealing  with  alternat- 
ing-current phenomena  ;  and  in  this  regard  the  book  may  be 
considered  as  an  expansion  and  extension  of  my  paper  on 
the  application  of  complex  imaginary  quantities  to  electri- 
cal engineering,  read  before  the  International  Electrical  Con- 


viii  PREFACE. 

gress  at  Chicago,  1893.  The  complex  imaginary  quantity 
is  gradually  introduced,  with  full  explanations,  the  algebraic 
operations  with  complex  quantities  being  discussed  in  Ap- 
pendix I.,  so  as  not  to  require  from  the  reader  any  previous 
knowledge  of  the  algebra  of  the  complex  imaginary  plane. 

While  those  phenomena  which  are  characteristic  to  poly- 
phase systems,  as  the  resultant  action  of  the  phases,  the 
effects  of  unbalancing,  the  transformation  of  polyphase  sys- 
tems, etc.,  have  been  discussed  separately  in  the  last  chap- 
ters, many  of  the  investigations  in  the  previous  parts  of  the 
book  apply  to  polyphase  systems  as  well  as  single-phase 
circuits,  as  the  chapters  on  induction  motors,  generators, 
synchronous  motors,  etc. 

A  part  of  the  book  is  original  investigation,  either  pub- 
lished here  for  the  first  time,  or  collected  from  previous 
publications  and  more  fully  explained.  Other  parts  have 
been  published  before  by  other  investigators,  either  in  the 
same,  or  more  frequently  in  a  different  form. 

I  have,  however,  omitted  altogether  literary  references, 
for  the  reason  that  incomplete  references  would  be  worse 
than  none,  while  complete  references  would  entail  the  ex- 
penditure of  much  more  time  than  is  at  my  disposal,  with- 
out offering  sufficient  compensation  ;  since  I  believe  that 
the  reader  who  wants  information  on  some  phenomenon  or 
apparatus  is  more  interested  in  the  information  than  in 
knowing  who  first  investigated  the  phenomenon. 

Special  attention  has  been  given  to  supply  a  complete 
and  extensive  index  for  easy  reference,  and  to  render  the 
book  as  free  from  errors  as  possible.  Nevertheless,  it  prob- 
ably contains  some  errors,  typographical  and  otherwise ; 
and  I  will  be  obliged  to  any  reader  who  on  discovering  an 
error  or  an  apparent  error  will  notify  me. 

I  take  pleasure  here  in  expressing  my  thanks  to  Messrs. 
W.  D.  WEAVER,  A.  E.  KENNELLY,  and  TOWNSEND  WOL- 
COTT,  for  the  interest  they  have  taken  in  the  book  while  in 
the  course  of  publication,  as  well  as  for  the  valuable  assist- 


PREFACE.  IX 

ance  given  by  them  in  correcting  and  standardizing  the  no- 
tation to  conform  with  the  international  system,  and  numer- 
ous valuable  suggestions  regarding  desirable  improvements. 
Thanks  are  due  also  to  the  publishers,  who  have  spared 
no  effort  or  expense  to  make  the  book  as  creditable  as  pos- 
sible mechanically. 

CHARLES   PROTEUS  STEINMETZ. 
January,  1897. 


CONTENTS. 


CHAP.  I.    Introduction.  — 

§  1,  p.    1.     Fundamental  laws  of  continuous  current  circuits. 

§  2,  p.    2.     Impedance,  reactance,  effective  resistance. 

§  3,  p.    3.     Electro-magnetism  as  source  of  reactance. 

§  4,  p.     5.     Capacity  as  source  of  reactance. 

§  5,  p.    6.     Joule's  law  and  power  equation  of  alternating  circuit. 

§  6,  p.    6.     Fundamental    wave    and    higher    harmonics,    alternating 

waves  with  and  without  even  harmonics. 

§  7,  p.    9.     Alternating  waves  as  sine  waves. 

CHAP.  II.    Instantaneous  Values  and  Integral  Values.  — 
§     8,  p.  11.     Integral  values  of  wave. 
§     9,  p.  13.     Ratio  of  mean  to  maximum  to  effective  value  of  wave. 

CHAP.  III.    Law  of  Electro-magnetic  Induction. — 
§  11,  p.  16.     Induced  E.M.F.  mean  value. 
§  12,  p.  17.     Induced  E.M.F.  effective  value. 
§  13,  p.  18.     Inductance  and  reactance. 

CHAP.  IV.    Graphic  Representation. — 

§  14,  p.  19.     Polar  characteristic  of  alternating  wave. 

§  15,  p.  20.     Polar  characteristic  of  sine  wave. 

§  16,  p.  21.     Parallelogram  of  sine  waves,  Kirchhoff's  laws,  and  energy 

equation. 

§  17,  p.  23.     Non-inductive  circuit  fed  over  inductive  line,  instance. 
§  18,  p.  24.     Counter  E.M.F.  and  component  of  impressed  E.M.F. 
§  19,  p.  26.     Continued. 
§  20,  p  26.     Inductive  circuit  and  circuit  with  leading  current  fed  over 

inductive  line.     Alternating-current  generator. 

§  21,  p.  28.     Polar  diagram  of  alternating-current  transformer,  instance. 
§  22,  p.  30.     Continued. 

CHAP.  V.    Symbolic  Method.— 

§  23,  p.  33.      Disadvantage  of  graphic  method  for  numerical  calculatioa 

§  24,  p.  34.     Trigonometric  calculation. 

§  25,  p.  34.      Rectangular  components  of  vectors. 

§  26,  p.  36.     Introduction  of  /  as  distinguishing  index. 

§  27,  p.  36.     Rotation  of  vector  by  180°  and  90°.  j  =  V^HT. 


xii  CONTENTS. 

CHAP.  V.    Symbolic  Method  —  Continued.  — 

§  28,  p.  37.     Combination  of  sine  waves  in  symbolic  expression. 

§  29,  p.  38.     Resistance,  reactance,  impedance,  in  symbolic  expression. 

§  30,  p.  40.     Capacity  reactance  in  symbolic  representation. 

§  31,  p.  40.     KirchhofF s  laws  in  symbolic  representation. 

§  32,  p.  41.     Circuit  supplied  over  inductive  line,  instance. 

CHAP.  VI.     Topographic  Method. — 
§  33,  p,  43.     Ambiguity  of  vectors. 
§  34,  p.  44.     Instance  of  a  three-phase  system. 
§  35,  p.  46.     Three-phase  generator  on  balanced  load. 
§  36,  p.  47.     Cable  with  distributed  capacity  and  resistance. 
§  37,  p.  49.     Transmission  line  with  self-inductive  capacity,  resistance, 
and  leakage. 

CHAP.  VII.     Admittance,  Conductance,  Susceptance.— 

§  38,  p.  52.     Combination  of  resistances  and  conductances  in  series  and 

in  parallel. 
§  39,  p.  53.     Combination    of   impedances.     Admittance,  conductance, 

susceptance. 
§  40,  p.  54.     Relation   between   impedance,   resistance,  reactance,  and 

admittance,  conductance,  susceptance. 
§  41,  p.  56.     Dependence  of  admittance,  conductance,  susceptance,  upon 

resistance  and  reactance.     Combination  of  impedances  and  ad- 

mittances. 

CHAP.  VIII.     Circuits    containing     Resistance,    Inductance,    and    Ca- 
pacity. — 

§  42,  p.  58.     Introduction. 
§  43,  p.  58.     Resistance  in  series  with  circuit. 
§  44,  p.  60.     Discussion  of  instances. 
§  45,  p.  61.     Reactance  in  series  with  circuit. 
§  46,  p.  64.     Discussion  of  instances. 
§  47,  p.  66.     Reactance  in  series  with  circuit. 
§  48,  p.  68.     Impedance  in  series  with  circuit. 
§  49,  p.  69.     Continued. 
§  50,  p.  71.     Instance. 

§  51,  p.  72.     Compensation  for  lagging  currents  by  shunted  condensance. 
§  52,  p.  73.     Complete  balance  by  variation  of  shunted  condensance. 
§  53,  p.  75.     Partial  balance  by  constant  shunted  condensance. 
§  54,  p.  76.     Constant  potential  —  constant  current  transformation. 
§  55,  p.  79.     Constant  current  —  constant  potential  transformation. 
§  56,  p.  81.     Efficiency  of  constant  potential  —  constant  current  trans- 
formation. 

CHAP.  IX.    Resistance  and  Reactance  of  Transmission  Lines.  — 
§  57,  p.  83.     Introduction. 
§  58,  p.  84.     Non-inductive  receiver  circuit  supplied  over  inductive  line. 


CONTENTS.  xiii 

CHAP.  IX.    Resistance  and  Reactance  of  Transmission  Lines. — Continued. 

§  59,  p.    86.     Instance. 

§  60,  p.    87.     Maximum  power  supplied  over  inductive  line. 

§  61,  p.  88.  Dependence  of  output  upon  the  susceptance  of  the  re- 
ceiver circuit. 

§  62,  p.  89.  Dependence  of  output  upon  the  conductance  of  the  re- 
ceiver circuit. 

§  63,  p.    90.     Summary. 

§  64,  p.    92.     Instance. 

§  65,  p.    93.     Condition  of  maximum  efficiency. 

§  €6,  p.    96.     Control  of  receiver  voltage  by  shunted  susceptance. 

§  67,  p.    97.     Compensation  for  line  drop  by  shunted  susceptance. 

§  68,  p.    97.     Maximum  output  and  discussion. 

§  69,  p.    98.     Instances. 

§  70,  p.  101.     Maxium  rise  of  potential  in  receiver  circuit. 

§  71,  p.  102.     Summary  and  instances. 

CHAP.  X.    Effective  Resistance  and  Reactance.  — 

§  72,  p.  104.  Effective  resistance,  reactance,  conductance,  and  suscep- 
tance. 

§  73,  p.  105.     Sources  of  energy  losses  in  alternating-current  circuits. 

§  74,  p.  106.     Magnetic  hysteresis. 

§  75,  p.  107.     Hysteretic  cycles  and  corresponding  current  waves. 

§  76,  p.  111.  Action  of  air-gap  and  of  induced  current  on  hysteretic 
distortion. 

§  77,  p.  111.     Equivalent  sine  wave  and  wattless  higher  harmonic. 

§  78,  p.  113.     True  and  apparent  magnetic  characteristic. 

§  79,  p.  115.     Angle  of  hysteretic  advance  of  phase. 

§  80,  p.  116.     Loss  of  energy  by  molecular  magnetic  friction. 

§  81,  p.  119.     Effective  conductance,  due  to  magnetic  hysteresis. 

§  82,  p.  122.  Absolute  admittance  of  ironclad  circuits  and  angle  of 
hysteretic  advance. 

§  83,  p.  124.      Magnetic  circuit  containing  air-gap. 

§  84,  p.  125.     Electric  constants  of  circuit  containing  iron. 

§  85,  p.  127.     Conclusion. 

CHAP.  XI.     Foucault  or  Eddy  Currents. — 

§  86,  p.  129.     Effective  conductance  of  eddy  currents. 

§  87,  p.  130.     Advance  angle  of  eddy  currents. 

§  88,  p.  131.     Loss  of  power  by  eddy  currents,  and  coefficient  of  eddy 

currents. 

§  89,  p.  131.     Laminated  iron. 
§  90,  p.  133.      Iron  wire. 

§  91,  p.  135.     Comparison  of  sheet  iron  and  iron  wire. 
§  92,  p.  136.     Demagnetizing  or  screening  effect  of  eddy  currents. 
§  93,  p.  138.     Continued. 
§  94,  p.  138.     Large  eddy  currents. 


CONTENTS. 

CHAP.  XI.    Foucault  or  Eddy  Currents.  —  Continued.  — 

§  95,  p.  139.     Eddy   currents  in  conductor  and   unequal  current  dis- 
tribution. 

§  96,  p.  140.     Continued. 
§  97,  p.  142.     Mutual  inductance. 
§  98,  p.  144.     Dielectric  and  electrostatic  phenomena. 
§  99,  p.  145.     Dielectric  hysteretic  admittance,  impedance,  lag,  etc. 
§  100,  p.  147.     Electrostatic  induction  or  influence. 
§  101,  p.  149.     Energy  components  and  wattless  components. 

CHAP.  XII.    Power,  and  Double  Frequency  Quantities  in  General. 
§  102,  p.  150.     Double  frequency  of  power. 
§  103,  p.  151.     Symbolic  representation  of  power. 
§  104,  p.  153.     Extra-algebraic  features  thereof. 
§  105,  p.  155.      Combination  of  powers. 
§  106,  p.  156.     Torque  as  double  frequency  product. 

CHAP.  XIII.    Distributed  Capacity,  Inductance,   Resistance,  and  Leak- 
age.— 

§  107,  p.  158.     Introduction. 

§  108,  p.  159.     Magnitude  of  charging  current  of  transmission  lines. 

§  109,  p.  160.  Line  capacity  represented  by  one  condenser  shunted 
across  middle  of  line. 

§  110,  p.  161.     Line  capacity  represented  by  three  condensers. 

§  111,  p.  163.  Complete  investigation  of  distributed  capacity,  induc- 
tance, leakage,  and  resistance. 

§  112,  p.  165.     Continued. 

§  113,  p.  166.     Continued. 

§  114,  p.  166.     Continued. 

§  115,  p.  167.     Continued. 

§  116,  p.  169.     Continued. 

§  117,  p.  170.     Continued. 

§  118,  p.  170.     Difference  of  phase  at  any  point  of  line. 

§  119,  p.  17-2.     Instance. 

§  120,  p.  173.     Further  instance  and  discussion. 

§  121,  p.  178.  Particular  cases,  open  circuit  at  end  of  line,  line 
grounded  at  end,  infinitely  ong  conductor,  generator  feeding 
into  closed  circuit. 

§  122,  p.  181.     Natural  period  of  transmission  line. 

§  123,  p.  186.     Discussion. 

§  124,  p.  190.     Continued. 

§  125,  p.  191.     Inductance  of  uniformly  charged  line. 

CHAP.  XIV.    The  Alternating-Current  Transformer.— 

§  126,  p.  193.  General. 

§  127,  p.  193.  Mutual  inductance  and  self-inductance  of  transformer. 

§  128,  p.  194.  Magnetic  circuit  of  transformer. 


CONTENTS. 


.  XV 


CHAP.  XIV.    The  Alternating-Current  Transformer  —  Continued.  — 
§  129,  p.  195.     Continued. 
§  130,  p.  196.     Polar  diagram  of  transformer. 
§  131,  p.  198.     Instance. 
§  132,  p.  202.     Diagram  for  varying  load. 
§  133,  p.  203.     Instance. 


134,  p.  204.     Symbolic  method,  equations. 


§  135,  p.  206. 
§  136,  p.  208. 


Continued. 

Apparent    impedance    of    transformer. 


Transformer 


equivalent  to  divided  circuit. 

§  137,  p.  209.  Continued. 

§  138,  p.  212.  Transformer  on  non-inductive  load. 

§  139,  p.  214.  Constants  of  transformer  on  non-inductive  load. 

§  140,  p.  217.  Numerical  instance. 

CHAP.  XV.     General    Alternating-Current     Transformer    or    Frequency 

Converters.  — 

§  141,  p.  219.  Introduction. 

§  142,  p.  220.  Magnetic  cross-flux  or  self-induction  of  transformer. 

§  143,  p.  221.  Mutual  flux  of  transformer. 

§  144,  p.  221.  Difference  of  frequency  between  primary  and  secondary 

of  general  alternate-current  transformer. 

§  145,  p.  221.  Equations  of  general  alternate-current  transformer. 

§  146,  p.  227.  Power,  output,  and  input,  mechanical  and  electrical. 

§  147,  p.  228.  Continued. 

§  148,  p.  229.  Speed  and  output. 

§  149,  p.  231.  Numerical  instance. 

§  150,  p.  232.  Characteristic  curves  of  frequency  converter. 

CHAP.  XVI.     Induction  Machines.— 

§  151,  p.  237.  Slip  and  secondary  frequency. 

§  152,  p.  238.  Equations  of  induction  motor. 

§  153,  p.  239.  Magnetic  flux,  admittance,  and  impedance. 

§  154,  p.  241.  E.M.F. 

§  155,  p.  244.  Graphic  representation. 

§  156,  p.  245.  Continued. 

§  157,  p.  246.  Torque  and  power. 

§  158,  p.  248.  Power  of  induction  motors. 

§  159,  p.  250.  Maximum  torque. 

§  160,  p.  252.  Continued. 

§  161,  p.  252.  Maximum  power. 

§  162,  p.  254.  Starting  torque. 

§  163,  p.  258.  Synchronism. 

§  164,  p.  258.  Near  synchronism. 

§  165,  p.  259.  Numerical  instance  of  induction  motor. 

§  166,  p.  262.  Calculation  of  induction  motor  curves. 

§  167,  p.  265.  Numerical  instance. 


xvi  CONTENTS. 

CHAP.  XVI.    Induction  Machines  —Continued. — 

§  168,  p.  265.  Induction  generator. 

§  169,  p.  268.  Power  factor  of  induction  generator. 

§  170,  p.  269.  Constant  speed,  induction  generator. 

§  171,  p.  272.  Induction  generator  and  synchronous  motor. 

§  172,  p.  274.  Concatenation  or  tandem  control  of  induction  motors. 

§  173,  p.  276.  Calculation  of  concatenated  couple. 

§  174,  p.  280.  Numerical  instance. 

§  175,  p.  281.  Single-phase  induction  motor. 

§  176,  p.  283.  Starting  devices  of  single-phase  motor. 

§  177,  p.  284.  Polyphase  motor  on  single-phase  circuit. 

§  178,  p.  286.  Condenser  in  tertiary  circuit. 

§  179,  p.  287.  Speed  curves  with  condenser. 

§  180,  p.  291.  Synchronous  induction  motor. 

§  181,  p.  293.  Hysteresis  motor. 

CHAP.  XVII.    Alternate-Current  Generator. — 

§  182,  p.  297.  Magnetic  reaction  of  lag  and  lead. 

§  183,  p.  300.  Self-inductance  and  synchronous  reactance. 

§  184,  p.  302.  Equations  of  alternator. 

§  185,  p.  303.  Numerical  instance,  field  characteristic. 

§  186,  p.  307.  Dependence  of  terminal  voltage  on  phase  relation. 

§  187,  p.  307.  Constant  potential  regulation. 

§  188,  p.  309.  Constant  current  regulation,  maximum  output. 

CHAP.  XVIII.     Synchronizing  Alternators.  — 

§  189,  p.  311.  Introduction. 

§  190,  p.  311.  Rigid  mechanical  connection. 

§  191,  p.  311.  Uniformity  of  speed 

§  192,  p.  312.  Synchronizing. 

§  193,  p.  313.  Running  in  synchronism. 

§  194,  p.  313.  Series  operation  of  alternators. 

§  195,  p.  314.  Equations  of  synchronous  running  alternators,  synchro- 
nizing power. 

§  196,  p.  317.  Special  case  of  equal  alternators  at  equal  excitation. 

§  197,  p.  320.  Numerical  instance. 

CHAP.  XIX.    Synchronous  Motor. — 

§  198,  p.  321.  Graphic  method. 

§  199,  p.  323.  Continued. 

§  200,  p.  325.  Instance. 

§  201,  p.  326.  Constant  impressed  E.M.F.  and  constant  current. 

§  202,  p.  329.  Constant  impressed  and  counter  E.M.F. 

§  203,  p.  332.  Constant  impressed  E.M.F.  and  maximum  efficiency. 

§  204,  p.  334.  Constant  impressed  E.M.F.  and  constant  output. 

§  205,  p.  338.  Analytical  method.     Fundamental  equations  and  power, 
characteristic. 


CONTENTS.  xvii 

CHAP.  XIX.    Synchronous  Motor  —  Continued. — 

§  206,  p.  342.  Maximum  output. 

§  207,  p.  343.  No  load. 

§  208,  p.  345.  Minimum  current. 

§  209,  p.  347.  Maximum  displacement  of  phase. 

§  210,  p.  349.  Constant  counter  E.M.F. 

§  211,  p.  349.  Numerical  instance. 

§  212,  p.  351.  Discussion  of  results. 

CHAP.  XX.     Commutator  Motors. — 

§  213,  p.  354.  Types  of  commutator  motors. 

§  214,  p.  354.  Repulsion  motor  as  induction  motor. 

§  215,  p.  356.  Two  types  of  repulsion  motors. 

§  216,  p.  358.  Definition  of  repulsion  motor. 

§  217,  p.  359.  Equations  of  repulsion  motor. 

§  218,  p.  360.  Continued. 

§  219,  p.  361.  Power  of  repulsion  motor.     Instance. 

§  220,  p.  363.  Series  motor,  shunt  motor. 

§  221,  p.  366.  Equations  of  series  motor. 

§  222,  p.  367.  Numerical  instance. 

§  223,  p.  368.  Shunt  motor. 

§  224,  p.  370.  Power  factor  of  series  motor. 

CHAP.  XXI.     Reaction  Machines. — 

§  225,  p.  371.  General  discussion. 

§  226,  p.  372.  Energy  component  of  reactance. 

§  227,  p.  372.  Hysteretic  energy  component  of  reactance. 

§  228,  p.  373.  Periodic  variation  reactance. 

§  229,  p.  375.  Distortion  of  wave-shape. 

§  230,  p.  377.  Unsymmetrical  distortion  of  wave-shape. 

§  231,  p.  378.  Equations  of  reaction  machines. 

§  232,  p.  380.  Numerical  instance. 

CHAP.  XXII.     Distortion  of  Wave-shape,  and  its  Causes.  — 

§  233,  p.  383.  Equivalent  sine  wave. 

§  234,  p.  383.  Cause  of  distortion. 

§  235,  p.  384.  Lack  of  uniformity  and  pulsation  of  magnetic  field^ 

S  236,  p.  388.  Continued. 

§  237,  p.  391.  Pulsation  of  reactance. 

§  238,  p.  391.  Pulsation  of  reactance  in  reaction  machine. 

§  239,  p.  393.  General  discussion. 

£  240,  p.  393.  Pulsation  of  resistance  arc. 

§  241,  p.  395.  Instance. 

§  242,  p.  396.  Distortion  of  wave-shape  by  arc. 

§  243.  p.  397.  Discussion. 


xvili  CO  TTENTS. 

CHAP.  XXIII.    Effects  of  Higher  Harmonics.— 

§  244,  p.  393.  Distortion  of  wave-shape  by  triple  and  quintuple  har- 
monics. Some  characteristic  wave-shapes. 

§  245,  p.  401.  Effect  of  self-induction  and  capacity  on  higher  harmonics. 

§  246,  p.  402.  Resonance  due  to  higher  harmonics  in  transmission  lines. 

§  247,  p.  405.  Power  of  complex  harmonic  waves. 

§  248,  p.  405.  Three-phase  generator. 

§  249,  p.  407.  Decrease  of  hysteresis  by  distortion  of  wave-shape. 

§  250,  p.  407.  Increase  of  hysteresis  by  distortion  of  wave-shape. 

§  251,  p.  408.  Eddy  currents. 

§  252,  p.  408.  Effect  of  distorted  waves  on  insulation. 

CHAP.  XXIV.     Symbolic  Representation  of  General  Alternating  Wave.  — 

§  253,  p.  410.  Symbolic  representation. 

§  254,  p.  412.  Effective  values. 

§  255,  p.  4l3.  Power  torque,  etc.     Circuit  factor. 

§  256,  p.  416.  Resistance,  inductance,  and  capacity  in  series. 

§  257,  p.  419.  Apparent  capacity  of  condenser. 

§  258,  p.  422.  Synchronous  motor. 

§  259,  p.  426.  Induction  motor. 

CHAP.  XXV.    General  Polyphase  Systems.— 

§  260,  p.  430.     Definition  of   systems,  symmetrical  and  unsymmetrical 

systems. 
§  261,  p.  430.     Flow   of   power.      Balanced   and    unbalanced   systems. 

Independent  and  interlinked  systems.     Star  connection  and  ring 

connection. 
§  262,  p.  432.     Classification  of  polyphase  systems. 

CHAP.  XXVI.     Symmetrical  Polyphase  Systems.— 

§  263,  p.  434.     General  equations  of  symmetrical  systems. 
§  264,  p.  435.     Particular  systems. 

§  265,  p.  436.     Resultant  M.M.F.  of  symmetrical  system. 
§  266,  p.  439.     Particular  systems. 

CHAP.  XXVII.     Balanced  and  Uunbalanced  Polyphase  Systems.  — 

§  267,  p.  440.  Flow  of  power  in  single-phase  system. 

§  268,  p.  441.  Flow  of  power  in  polyphase  systems,  balance  factor  of 

system. 

§  269,  p.  442.  Balance  factor. 

§  270,  p.  442.  Three-phase  system,  quarter-phase  system. 

§  271,  p.  413.  Inverted  three  phase  system. 

§  272,  p.  444.  Diagrams  of  flow  of  power. 

§  273,  p.  447.  Monocyclic  and  polycyclic  systems. 

§  274,  p.  447.  Power  characteristic  of  alternating-current  system. 

§  275,  p.  448.  The  same  in  rectangular  coordinates. 

§  276,  p.  450.  Main  power  axes  of  alternating-current  system. 


CONTENTS.  XIX 

CHAP.  XXVIII.    Interlinked  Polyphase  Systems.— 

§  277,  p.  452.     Interlinked  and  independent  systems. 

§  278,  p.  452.  Star  connection  and  ring  connection.  Y  connection  and 
delta  connection. 

§  279,  p.  454.      Continued. 

§  280,  p.  455.  Star  potential  and  ring  potential.  Star  current  and  ring 
current.  Y  potential  and  Y  current,  delta  potential  and  delta 
current. 

§  281,  p.  455.      Equations  of  interlinked  polyphase  systems. 

§  282,  p.  457.      Continued. 
CHAP.  XXIX.     Transformation  of  Polyphase  Systems. — 

§  283,  p.  460.      Constancy  of  balance  factor. 

§  284,  p.  460.      Equations  of  transformation  of  polyphase  systems. 

§  285,  p.  462.     Three-phase,  quarter-phase  transformation. 

§  286,  p.  463.     Some  of  the  more  common  polyphase  transformations. 

§  287,  p.  466.f     Transformation  with  change  of  balance  factor. 

CHAP.  XXX.     Copper  Efficiency  of  Systems. — 

§  288,  p.  468.     General  discussion. 

§  289,  p.  469.  Comparison  on  the  basis  of  equality  of  minimum  dif- 
ference of  potential. 

§  290,  p.  474.  Comparison  on  the  basis  of  equality  of  maximum  dif- 
ference of  potential. 

§  291,  p.  476.      Continued. 

CHAP.  XXXI.     Three-phase  System.— 
§  292,  p.  478.     General  equations. 

§  293,  p.  481.  Special  cases:  balanced  system,  one  branch  loaded, 
two  branches  loaded. 

CHAP.  XXXII.     Quarter-phase  System. — 
§  294,  p.  483.     General  equations. 
§  295,  p.  484.     Special  cases  :  balanced  system,  one  branch  loaded. 

APPENDIX  I.     Algebra  of  Complex  Imaginary  Quantities.— 

§  296,  p.  489.  Introduction. 

§  297,  p.  489.  Numeration,  addition,  multiplication,  involution. 

§  298,  p.  490.  Subtraction,  negative  number. 

§  299,  p.  491.  Division,  fraction. 

§  300,  p.  491.  Evolution  and  logarithmation. 

§  301,  p.  492.  Imaginary  unit,  complex  imaginary  number. 

§  302,  p.  492.  Review. 

§  303,  p.  493.  Algebraic  operations  with  complex  quantities. 

§  304,  p.  494.  Continued. 

§  305,  p.  495.  Roots  of  the  unit. 

§  306,  p.  495.  Rotation. 

§  307,  p.  496.  Complex  imaginary  plane. 


CONTENTS. 


APPENDIX  II.    Oscillating  Currents. — 

§  308,  p.  497.  Introduction. 

§  309,  p.  498.  General  equations. 

§  310,  p.  499.  Polar  coordinates. 

§  311,  p.  500.  Loxodromic  spiral. 

§  312,  p.  501.  Impedance  and  admittance. 

§  313,  p.  502.  Inductance. 

§  314,  p.  502.  Capacity. 

§  315,  p.  503.  Impedance. 

§  316,  p.  504.  Admittance. 

§  317,  p.  505.  Conductance  and  susceptance. 

§  318,  p.  506.  Circuits  of  zero  impedance. 

§  319,  p.  506.  Continued. 

§  320,  p.  507.  Origin  of  oscillating  currents. 

§  321,  p.  508.  Oscillating  discharge. 

§  322,  p.  509.  Oscillating  discharge  of  condensers 

§  323,  p.  510.  Oscillating  current  transformer. 

§  324,  p.  512.  Fundamental  equations  thereof. 


THEORY    AND    CALCULATION 

OF 

ALTERNATING-CURRENT   PHENOMENA. 


CHAPTER    I. 

INTRODUCTION. 

1.  IN  the  practical  applications  of  electrical  energy,  we 
meet  with  two  different  classes  of  phenomena,  due  respec- 
tively to  the  continuous  current  and  to  the  alternating 
current. 

The  continuous-current  phenomena  have  been  brought 
within  the  realm  of  exact  analytical  calculation  by  a  few 
fundamental  laws  :  — 

1.)  Ohm's  law  :  i  =  e  j  r,  where  r,  the  resistance,  is  a 
constant  of  the  circuit. 

2.)  Joule's  law:  P=  izr,  where  P  is  the  rate  at  which 
energy  is  expended  by  the  current,  i,  in  the  resistance,  r. 

3.)  The  power  equation  :  P0  =  ei,  where  P0  is  the 
power  expended  in  the  circuit  of  E.M.F.,  e,  and  current,  /. 

4.)    Kirchhoff's  laws  : 

a.}  The  sum  of  all  the  E.M.Fs.  in  a  closed  circuit  =  0, 
if  the  E.M.F.  consumed  by  the  resistance,  ir,  is  also  con- 
sidered as  a  counter  E.M.F.,  and  all  the  E.M.Fs.  are  taken 
in  their  proper  direction. 

b.)  The  sum  of  all  the  currents  flowing  towards  a  dis- 
tributing point  =  0. 

In  alternating-current  circuits,  that  is,  in  circuits  con- 
veying curr'ents  which  rapidly  and  periodically  change  their 


2  ALTERNATING-CURRENT  PHENOMENA. 

direction,  these  laws  cease  to  hold.  Energy  is  expended, 
not  only  in  the  conductor  through  its  ohmic  resistance,  but 
also  outside  of  it ;  energy  is  stored  up  and  returned,  so 
that  large  currents  may  flow,  impressed  by  high  E.M.Fs., 
without  representing  any  considerable  amount  of  expended 
energy,  but  merely  a  surging  to  and  fro  of  energy ;  the 
ohmic  resistance  ceases  to  be  the  determining  factor  of 
current  strength ;  currents  may  divide  into  components, 
each  of  which  is  larger  than  the  undivided  current,  etc. 

2.  In  place  of  the  above-mentioned  fundamental  laws  of 
continuous  currents,  we  find  in  alternating-current  circuits 
the  following  : 

Ohm's  law  assumes  the  form,  i  =  e ]  s,  where  z,  the 
apparent  resistance,  or  impedance,  is  no  longer  a  constant 
of  the  circuit,  but  depends  upon  the  frequency  of  the  cur- 
rents ;  and  in  circuits  containing  iron,  etc.,  also  upon  the 
E.M.F. 

Impedance,  z,  is,  in  the  system  of  absolute  units,  of  the 
same  dimensions  as  resistance  (that  is,  of  the  dimension 
LT~l  =  velocity),  and  is  expressed  in  ohms. 

It  consists  of  two  components,  the  resistance,  r,  and  the 

reactance,  x,  or  —  , 

0=  Vr2  +  Ar2. 

The  resistance,  r,  in  circuits  where  energy  is  expended 
only  in  heating  the  conductor,  is  the  same  as  the  ohmic 
resistance  of  continuous-current  circuits.  In  circuits,  how- 
ever, where  energy  is  also  expended  outside  of  the  con- 
ductor by  magnetic  hysteresis,  mutual  inductance,  dielectric 
hysteresis,  etc.,  r  is  larger  than  the  true  ohmic  resistance 
of  the  conductor,  since  it  refers  to  the  total  expenditure  of 
energy.  It  may  be  called  then  the  effective  resistance.  It 
is  no  longer  a  constant  of  the  circuit. 

The  reactance,  x,  does  not  represent  the  expenditure  of 
power,  as  does  the  effective  resistance,  r,  but  merely  the 
surging  to  and  fro  of  energy.  It  is  not  a  constant  of  the 


INTRODUCTION.  3 

circuit,  but  depends  upon  the  frequency,  and  frequently, 
as  in  circuits  containing  iron,  or  in  electrolytic  conductors, 
upon  the  E.M.F.  also.  Hence,  while  the  effective  resist- 
ance, r,  refers  to  the  energy  component  of  E.M.F.,  or  the 
E.M.F.  in  phase  with  the  current,  the  reactance,  x,  refers 
to  the  wattless  component  of  E.M.F.,  or  the  E.M.F.  in 
quadrature  with  the  current. 

3.  The  principal  sources  of  reactance  are  electro-mag- 
netism and  capacity. 

ELECTRO— MAGNETISM. 

An  electric  current,  i,  flowing  through  a  circuit,  produces 
a  magnetic  flux  surrounding  the  conductor  in  lines  of 
magnetic  force  (or  more  correctly,  lines  of  magnetic  induc- 
tion), of  closed,  circular,  or  other  form,  which  alternate 
with  the  alternations  of  the  current,  and  thereby  induce 
an  E.M.F.  in  the  conductor.  Since  the  magnetic  flux  is 
in  phase  with  the  current,  and  the  induced  E.M.F.  90°,  or 
a  quarter  period,  behind  the  flux,  this  E.M.F.  of  self -induc- 
tance lags  90°,  or  a  quarter  period,  behind  the  current ;  that 
is,  is  in  quadrature  therewith,  and  therefore  wattless. 

If  now  4>  =  the  magnetic  flux  produced  by,  and  inter- 
linked  with,  the  current  i  (where  those  lines  of  magnetic 
force,  which  are  interlinked  w-fold,  or  pass  around  n  turns 
of  the  conductor,  are  counted  n  times),  the  ratio,  $  /  z,  is 
denoted  by  L,  and  called  self -inductance,  or  the  coefficient  of 
self-induction  of  the  circuit.  It  is  numerically  equal,  in 
absolute  units,  to  the  interlinkages  of  the  circuit  with  the 
magnetic  flux  produced  by  unit  current,  and  is,  in  the 
system  of  absolute  units,  of  the  dimension  of  length.  In- 
stead of  the  self-inductance,  L,  sometimes  its  ratio  with 
the  ohmic  resistance,  r,  is  used,  and  is  called  the  Time- 
Constant  of  the  circuit  : 


4  ALTERNATING-CURRENT  PHENOMENA. 

If  a  conductor  surrounds  with  ;/  turns  a  magnetic  cir- 
cuit of  reluctance,  (R,  the  current,  i,  in  the  conductor  repre- 
sents the  M.M.F.  of  ni  ampere-turns,  and  hence  produces 
a  magnetic  flux  of  »//(R  lines  of  magnetic  force,  sur- 
rounding each  n  turns  of  the  conductor,  and  thereby  giving 
<1>  =:  ;/2//(R  interlinkages  between  the  magnetic  and  electric 
circuits.  Hence  the  inductance  is  L  =  $/  i  =  ;/2/(R. 

The  fundamental  law  of  electro-magnetic  induction  is, 
that  the  E.M.F.  induced  in  a  conductor  by  a  varying  mag- 
netic field  is  the  rate  of  cutting  of  the  conductor  through 
the  magnetic  field. 

Hence,  if  /  is  the  current,  and  L  is  the  inductance  of 
a  circuit,  the  magnetic  flux  interlinked  with  a  circuit  of 
current,  z,  is  Li,  and  4  NLi  is  consequently  the  average 
rate  of  cutting  ;  that  is,  the  number  of  lines  of  force  cut 
by  the  conductor  per  second,  where  N  '  =  frequency,  or 
number  of  complete  periods  (double  reversals)  of  the  cur- 
rent per  second. 

Since  the  maximum  rate  of  cutting  bears  to  the  average 
rate  the  same  ratio  as  the  quadrant  to  the  radius  of  a 
circle  (a  sinusoidal  variation  supposed),  that  is  the  ratio 
ir/2  H-  1,  the  maximum  rate  of  cutting  is  2-n-N,  and,  conse- 
quently, the  maximum  value  of  E.M.F.  induced  in  a  cir- 
cuit of  maximum  current  strength,  i,  and  inductance,  L,  is, 


Since  the  maximum  values  of  sine  waves  are  proportional 
(by  factor  V2)  to  the  effective  values  (square  root  of  mean 
squares),  if  i  =  effective  value  of  alternating  current,  e  = 
2  TT  NLi  is  the  effective  value  of  E.M.F.  of  self-inductance, 
and  the  ratio,  e  I  i  —  2  TT  NL,  is  the  magnetic  reactance  : 

xm  =  2  TT  NL. 
Thus,  \ir—  resistance,  xm  =  reactance,  z  =  impedance,— 

the  E.M.F.  consumed  by  resistance  is  :  el  =  ir  ; 
the  E.M.F.  consumed  by  reactance  is  :  <?2  =  /v/;,  : 


INTRODUCTION.  5 

and,  since  both  E.M.Fs.  are  in  quadrature  to  each  other, 
the  total  E.M.F.  is  — 


e 

that  is,  the  impedance,  z,  takes  in  alternating-current  cir- 
cuits the  place  of  the  resistance,  r,  in  continuous-current 
circuits. 

CAPACITY. 

4.  If  upon  a  condenser  of  capacity,  C,  an  E.M.F.,  e,  is 
impressed,  the  condenser  receives  the  electrostatic  charge,  Ce. 

If  the  E.M.F.,  e,  alternates  with  the  frequency,  N,  the 
average  rate  of  charge  and  discharge  is  4  IV,  and  2  TT  N  the 
maximum  rate  of  charge  and  discharge,  sinusoidal  waves  sup- 
posed, hence,  i  —  2  TT  ./VCV  the  current  passing  into  the  con- 
denser, which  is  in  quadrature  to  the  E.M.F.,  and  leading. 


It  is  then:- 


the  "capacity  reactance"  or  "  condensance" 

Polarization  in  electrolytic  conductors  acts  to  a  certain 
extent  like  capacity. 

The  capacity  reactance  is  inversely  proportional  to  the 
frequency,  and  represents  the  leading  out-of  -phase  wave; 
the  magnetic  reactance  is  directly  proportional  to  the 
frequency,  and  represents  the  lagging  out-of-phase  wave. 
Hence  both  are  of  opposite  sign  with  regard  to  each  other, 
and  the  total  reactance  of  the  circuit  is  their  difference, 

*  '  =  Xm  -•**• 

The  total  resistance  of  a  circuit  is  equal  to  the  sum  of 
all  the  resistances  connected  in  series  ;  the  total  reactance 
of  a  circuit  is  equal  to  the  algebraic  sum  of  all  the  reac- 
tances connected  in  series  ;  the  total  impedance  of  a  circuit, 
however,  is  not  equal  to  the  sum  of  all  the  individual 
impedances,  but  in  general  less,  and  is  the  resultant  of  the 
total  resistance  and  the  total  reactance.  Hence  it  is  not 
permissible  directly  to  add  impedances,  as  it  is  with  resist- 
ances or  reactances. 


6  AL  TERN  A  TIA'G-  CURRENT  PHENOMENA, 

A  further  discussion  cf  these  quantities  will  be  found  in 
the  later  chapters. 

5.  In   Joule's    law,   P  =  i2r,   r  is  not   the  true  ohmic 
resistance  any  more,  but  the  "  effective  resistance ; "  that 
is,  the  ratio  of  the  energy  component  of  E.M.F.  to  the  cur- 
rent.     Since  in  alternating-current  circuits,  besides  by  the 
ohmic    resistance    of   the    conductor,   energy   is    expended, 
partly    outside,    partly  even   inside,   of    the   conductor,   by 
magnetic  hysteresis,  mutual  inductance,  dielectric  hystere- 
sis, etc.,  the  effective  resistance,  r,  is  in  general  larger  than 
the  true  resistance  of  the  conductor,  sometimes  many  times 
larger,  as  in  transformers  at  open  secondary  circuit,  and  is 
not  a  constant  of  the  circuit  any  more.     It  is  more  fully 
discussed  in  Chapter  VII. 

In  alternating-current  circuits,  the  power  equation  con- 
tains a  third  term,  which,  in  sine  waves,  is  the  cosine  of 
the  difference  of  phase  between  E.M.F.  and  current :  — 

P0  =  ei  cos  <£. 

Consequently,  even  if  e  and  i  are  both  large,  P0  may  be 
very  small,  if  cos  <f>  is  small,  that  is,  <f>  near  90°. 

Kirchhoff's  laws  become  meaningless  in  their  original 
form,  since  these  laws  consider  the  E.M.Fs.  and  currents 
as  directional  quantities,  counted  positive  in  the  one,  nega- 
tive in  the  opposite  direction,  while  the  alternating  current 
has  no  definite  direction  of  its  own. 

6.  The   alternating  waves    may  have  widely  different 
shapes ;    some    of  the   more  frequent    ones   are   shown   in 
a  later  chapter. 

The  simplest  form,  however,  is  the  sine  wave,  shown  in 
Fig.  1,  or,  at  least,  a  wave  very  near  sine  shape,  which 
may  be  represented  analytically  by  :  — 

/  =  /  sin  ^  (/  -  4)  =  /sin  2  TT yV  (/  -  4)  ; 


INTRO  D  UC  TION. 


where  /  is  the  maximum  value  of  the  wave,  or  its  ampli- 
tude ;  T  is  the  time  of  one  complete  cyclic  repetition,  or 
the  period  of  the  wave,  or  N  =  1  /  T  is  the  frequency  or 
number  of  complete  periods  per  second  ;  and  t\  is  the  time, 
where  the  wave  is  zero,  or  the  epoch  of  the  wave,  generally 
called  the  pliasc* 

Obviously,  "phase"  or  "epoch"  attains  a  practical 
meaning  only  when  several  waves  of  different  phases  are 
considered,  as  "difference  of  phase."  When  dealing  with 
one  wave  only,  we  may  count  the  time  from  the  moment 


T\ 


rS 


Fig.  1.    Sine  Wave, 

where  the  wave  is  zero,  or  from  the  moment  of  its  maxi- 
mum, and  then  represent  it  by  :  — 

«  =  /  sin  2  TT  Nt  ; 
or,  /  =  /cos  2  TT  Nt. 

Since  it  is  a  univalent  function  of  time,  that  is,  can  at  a 
given  instant  have  one  value  only,  by  Fourier's  theorem, 
any  alternating  wave,  no  matter  what  its  shape  may  be, 
can  be  represented  by  a  series  of  sine  functions  of  different 
frequencies  and  different  phases,  in  the  form  :  — 

/  =  7i  sin  2  irN(t  —  A)  +  72  sin  4  TrJV(t  -  /2) 
+  73  sin 


*  "  Epoch  "  is  the  time  where  a  periodic  function  reaches  a  certain  value, 
for  instance,  zero;  and  "phase"  is  the  angular  position,  with  respect  to  a 
datum  position,  of  a  periodic  function  at  a  given  time.  Both  are  in  alternate- 
current  phenomena  only  different  ways  of  expressing  the  same  thing. 


8 


ALTERNA TING-CURRENT  PHENOMENA. 


where  fv  72,  73,  .  .  .  are  the  maximum  values  of  the  differ- 
ent components  of  the  wave,  fv  fv  /3  .  .  .  the  times,  where 
the  respective  components  pass  the  zero  value. 

The  first  term,  7X  sin  lir  N  (t  —  tj,  is  called  the  fun- 
damental wave,  or  the  first  harmonic;  the  further  terms  are 
called  the  higher  harmonics,  or  "overtones,"  in  analogy  to 
the  overtones  of  sound  waves.  In  sin  2  mr  N  (t  —  /„)  is  the 
«th  harmonic. 

By  resolving  the  sine  functions  of  the  time  differences, 
/  —  fp  t  —  /2  .  .  .  ,  we  reduce  the  general  expression  of 
the  wave  to  the  form  : 

Al  sin  2  TrNt  +  A*  sin  4  vNt  +  Az  sin  G  TT Nt  +  .  .  . 
1cos27rA?-f^2cos47rA?-f  ^8cos67ry\7+  .  .  . 


F/g.  2.     Wave  without  Even  Harmonics. 

The  two  half-waves  of  each  period,  the  positive  wave 
and  the  negative  wave  (counting  in  a  definite  direction  in 
the  circuit),  are  almost  always  identical.  Hence  the  even 
higher  harmonics,  which  cause  a  difference  in  the  shape  of 
the  two  half -waves,  disappear,  and  only  the  odd  harmonics 
exist,  except  in  very  special  cases. 

Hence  the  general  alternating-current  wave  is  expressed 

ty  :  i  =  7i  sin  2  TT  N(t  —  A)  +  7,  sin  6  TT  N  (t  —  /3) 

+  75  sin  10  TT  A^(/  —  /5)  +  ... 
or, 

/  =  ^  sin  2  TT  A7  +  Az  sin  6  TT  A7  +  A&  sin  10  w  A?  +  .  .  . 
cos  2  TT Nt  +  ^8  cos  6  TrNt  +  ^5  cos  10  vNt  +  .  .  . 


INTR  OD  UC  TION. 


9 


Such  a  wave  is  shown  in  Fig.  2,  while  Fig.  3  shows  a 
wave  whose  half-waves  are  different.  Figs.  2  and  3  repre- 
sent the  secondary  currents  of  a  Ruhmkorff  coil,  whose 
secondary  coil  is  closed  by  a  high  external  resistance  :  Fig. 
3  is  the  coil  operated  in  the  usual  way,  by  make  and  break 
of  the  primary  battery  current ;  Fig.  2  is  the  coil  fed  with 
reversed  currents  by  a  commutator  from  a  battery. 

7.  Self-inductance,  or  electro-magnetic  momentum,  which 
is  always  present  in  alternating-current  circuits,  —  to  a 
large  extent  in  generators,  transformers,  etc.,  —  tends  to 


Fig.  3.     Wave  with  Even  Harmonics. 

suppress  the  higher  harmonics  of  a  complex  harmonic  wave 
more  than  the  fundamental  harmonic,  since  the  self-induc- 
tive reactance  is  proportional  to  the  frequency,  and  is  thus 
greater  with  the  higher  harmonics,  and  thereby  causes  a 
general  tendency  towards  simple  sine  shape,  which  has  the 
effect,  that,  in  general,  the  alternating  currents  in  our  light 
and  power  circuits  are  sufficiently  near  sine  waves  to  make 
the  assumption  of  sine  shape  permissible. 

Hence,  in  the  calculation  of  alternating-current  phev 
nomena,  we  can  safely  assume  the  alternating  wave  as  a 
sine  wave,  without  making  any  serious  error  ;  and  it  will  be 


10  AL  TERN  A  TING-CURRENT  PHENOMENA. 

sufficient  to  keep  the  distortion  from  sine  shape  in  mind  as 
a  possible  disturbing  factor,  which  generally,  however,  is  in 
practice  negligible  —  perhaps  with  the  only  exception  of 
low-resistance  circuits  containing  large  magnetic  reactance, 
and  large  condensance  in  series  with  each  other,  so  as  to 
produce  resonance  effects  of  these  higher  harmonics. 


INSTANTANEOUS  AND  INTEGRAL    VALUES. 


11 


CHAPTER    II 

INSTANTANEOUS  VALUES  AND  INTEGRAL  VALUES. 

8.  IN  a  periodically  varying  function,  as  an  alternating 
current,  we  have  to  distinguish  between  the  instantaneous 
value,  which  varies  constantly  as  function  of  the  time,  and 
the  integral  value,  which  characterizes  the  wave  as  a  whole. 

As  such  integral  value,  almost  exclusively  the  effective 


Fig.  4.    Alternating  Wave. 

value  is  used,  that  is,  the  square  root  of  the  mean  squares  ; 
and  wherever  the  intensity  of  an  electric  wave  is  mentioned 
without  further  reference,  the  effective  value  is  understood. 

The  maximum  value  of  the  wave  is  of  practical  interest 
only  in  few  cases,  and  may,  besides,  be  different  for  the  two 
half-waves,  as  in  Fig.  3. 

As  arithmetic  mean,  or  average  value,  of  a  wave  as  in 
Figs.  4  and  5,  the  arithmetical  average  of  all  the  instan- 
taneous values  during  one  complete  period  is  understood. 

This  arithmetic  mean  is  either  =  0,  as  in  Fig.  4,  or  it 
differs  from  0,  as.  in  Fig.  5.  In  the  first  case,  the  wave 
is  called  an  alternating  wave,  in  the  latter  a  pttlsating  wave. 


12 


ALTERNA TING-CURRENT  PHENOMENA. 


Thus,  an  alternating  wave  is  a  wave  whose  positive 
values  give  the  same  sum  total  as  the  negative  values  ;  that 
is,  whose  two  half-waves  have  in  rectangular  coordinates 
the  same  area,  as  shown  in  Fig.  4. 

A  pulsating  wave  is  a  wave  in  which  one  of  the  half- 
waves  preponderates,  as  in  Fig.  5. 

By  electromagnetic  induction,  pulsating  waves  are  pro- 
duced only  by  commutating  and  unipolar  machines  (or  by 
the  superposition  of  alternating  upon  direct  currents,  etc.). 

All  inductive  apparatus  without  commutation  give  ex- 
clusively alternating  waves,  because,  no  matter  what  con- 


Fig.  5.    Pulsating  Wave. 

ditions  may  exist  in  the  circuit,  any  line  of  magnetic  force, 
which  during  a  complete  period  is  cut  by  the  circuit,  and 
thereby  induces  an  E.M.F.,  must  during  the  same  period 
be  cut  again  in  the  opposite  direction,  and  thereby  induce 
the  same  total  amount  of  E.M.F.  (Obviously,  this  does 
not  apply  to  circuits  consisting  of  different  parts  movable 
with  regard  to  each  other,  as  in  unipolar  machines.) 

In  the  following  we  shall  almost  exclusively  consider  the 
alternating  wave,  that  is  the  wave  whose  true  arithmetic 
mean  value  =  0. 

Frequently,  by  mean  value  of  an  alternating  wave,  the 
average  of  one  half-wave  only  is  denoted,  or  rather  the 


INSTANTANEOUS  AND  INTEGRAL    VALUES. 


13 


average  of  all  instantaneous  values  without  regard  to  their 
sign.  This  mean  value  is  of  no  practical  importance,  and 
is,  besides,  in  many  cases  indefinite. 

9.    In  a  sine  wave,  the  relation  of  the  mean  to  the  maxi- 
mum value  is  found  in  the  following  way :  — 


Fig.  8. 

Let,  in  Fig.  6,  AOB  represent  a  quadrant  of  a  circle 
with  radius  1. 

Then,  while  the  angle  <£  traverses  the  arc  -n-  /  2  from  A  to 
B,  the  sine  varies  from  0  to  OB  =  1.  Hence  the  average 
variation  of  the  sine  bears  to  that  of  the  corresponding  arc 
the  ratio  1  -j-  7r/2,  or  2  /  TT  •+-  1.  The  maximum  variation 
of  the  sine  takes  place  about  its  zero  value,  where  the  sine 
is  equal  to  the  arc.  Hence  the  maximum  variation  of  the 
sine  is  equal  to  the  variation  of  the  corresponding  arc,  and 
consequently  the  maximum  variation  of  the  sine  bears  to 
its  average  variation  the  same  ratio  as  the  average  variation 
of  the  arc  to  that  of  the  sine ;  that  is,  1  -f-  2  /  77-,  and  since 
the  variations  of  a  sine-function  are  sinusoidal  also,  we 
have, 

o 

Mean  value  of  sine  wave  -r-  maximum  value  =  • — •  -f-  1 

7T 

=  .63663. 

The  quantities,  "current,"  "E.M.F.,"  "magnetism,"  etc., 
are  in  reality  mathematical  fictions  only,  as  the  components 


14  AL  TERNA  TING-CURRENT  PHENOMENA. 

of  the  entities,  "energy,"  "power,"  etc. ;  that  is,  they  have 
no  independent  existence,  but  appear  only  as  squares  or 
products. 

Consequently,  the  only  integral  value  of  an  alternating 
wave  which  is  of  practical  importance,  as  directly  connected 
with  the  mechanical  system  of  units,  is  that  value  which 
represents  the  same  power  or  effect  as  the  periodical  wave. 
This  is  called  the  effective  value.  Its  square  is  equal  to  the 
mean  square  of  the  periodic  function,  that  is  :  — 

TJie  effective  value  of  an  alternating  wave,  or  tJie  value 
representing  the  same  effect  as  the  periodically  varying  wave, 
is  the  square  root  of  the  mean  square. 

In  a  sine  wave,  its  relation  to  the  maximum  value  is 
found  in  the  following  way  : 


Fig.  7. 


Let,  in  Fig.  7,  AOB  represent  a  quadrant  of  a  circle 
with  radius  1. 

Then,  since  the  sines  of  any  angle  </>  and  its  complemen- 
tary angle,  90°—  <£,  fulfill  the  condition,  — 

sin2  $  +  sin2  (90  —  <£)  =  1, 

the  sines  in  the  quadrant,  AOB,  can  be  grouped  into  pairs, 
so  that  the  sum  of  the  squares  of  any  pair  =  1  ;  or,  in  other 
words,  the  mean  square  of  the  sine  =1/2,  and  the  square 
root  of  the  mean  square,  or  the  effective  value  of  the  sine, 
=  1/V2.  That  is: 


INSTANTANEOUS  AND   INTEGRAL    VALUES. 


15 


The  effective  value  of  a  sine  function  bears  to  its 
mum  value  the  ratio,  — 
1 

V2 

Hence,  we  have  for  the  sine  curve  the  following  rela- 
tions : 


1  =  .70711. 


MAX. 

EFF. 

ARITH.  MEAN. 

Half 
Period. 

Whole 
Period. 

1 

1 

V2 

2 

7T 

0 

1 

.7071 

.63663 

0 

1.4142 

1 

.90034 

0 

1.5708 

1.1107 

1 

0 

10.    Coming  now  to  the  general  alternating  wave, 

/  =  Ai  sin  27r  Nt  +  Az  sin  4-n-  Nt  +  A3  sin  GTT  Nt  +  .  .  . 
+  BI  cos  2-n-Nt  +  B*  cos  ±TrNt  +  £s  cos  GTT  Nt  +  .  . 

we  find,  by  squaring  this  expression  and  canceling  all  the 
products  which  give  0  as  mean  square,  the  effective  value,  — 


1=  V*  W 

The  mean  value  does  not  give  a  simple  expression,  and 
is  of  no  general  interest. 


16  ALTERNATING-CURRENT  PHENOMENA, 


CHAPTER   III. 

LAW   OF    ELECTRO-MAGNETIC    INDUCTION. 

11.  If  an  electric  conductor  moves  relatively  to  a  mag- 
netic field,  an  E.M.F.  is  induced  in  the  conductor  which  is 
proportional  to  the  intensity  of  the  magnetic  field,  to  the 
length  of  the  conductor,  and  to  the  speed  of  its  motion 
perpendicular  to  the  magnetic  field  and  the  direction  of  the 
conductor ;  or,  in  other  words,  proportional  to  the  number 
of  lines  of  magnetic  force  cut  per  second  by  the  conductor. 

As  a  practical  unit  of  E.M.F.,  the  volt  is  defined  as  the 
E.M.F.  induced  in  a  conductor,  which  cuts  108  =  100,000,000 
lines  of  magnetic  force  per  second. 

If  the  conductor  is  closed  upon  itself,  the  induced  E.M.F. 
produces  a  current. 

A  closed  conductor  may  be  called  a  turn  or  a  convolution. 
In  such  a  turn,  the  number  of  lines  of  magnetic  force  cut 
per  second  is  the  increase  or  decrease  of  the  number  of 
lines  inclosed  by  the  turn,  or  n  times  as  large  with  n  turns. 

Hence  the  E.M.F.  in  volts  induced  in  n  turns,  or  con- 
volutions, is  n  times  the  increase  or  decrease,  per  second, 
of  the  flux  inclosed  by  the  turns,  times  10~8. 

If  the  change  of  the  flux  inclosed  by  the  turn,  or  by  n 
turns,  does  not  take  place  uniformly,  the  product  of  the 
number  of  turns,  times  change  of  flux  per  second,  gives 
the  average  E.M.F. 

If  the  magnetic  flux,  4>,  alternates  relatively  to  a  number 
of  turns,  n  —  that  is,  when  the  turns  either  revolve  through 
the  flux,  or  the  flux  passes  in  and  out  of  the  turns,  the  total 
flux  is  cut  four  times  during  each  complete  period  or  cycle, 
twice  passing  into,  and  twice  out  of,  the  turns. 


LAW  OF  ELECTRO-MAGNETIC  INDUCTION.  17 

Hence,  if  N=  number  of  complete  cycles  per  second, 
or  the  frequency  of  the  flux  3>,  the  average  E.M.F.  induced 
in  n  turns  is, 

£&vg,  =  4  «  3>  N  10  ~  8  volts. 

This  is  the  fundamental  equation  of  electrical  engineer- 
ing, and  applies  to  .continuous-current,  as  well  as  to  alter- 
nating-current, apparatus. 

12.  In  continuous-current  machines  and  in  many  alter- 
nators, the  turns  revolve  through  a  constant  magnetic 
field  ;  in  other  alternators  and  in  induction  motors,  the  mag- 
netic field  revolves  ;  in  transformers,  the  field  alternates 
with  respect  to  the  stationary  turns. 

Thus,  in  the  continuous-current  machine,  if  n  =  num- 
ber of  turns  in  series  from  brush  to  brush,  <I>  =  flux  inclosed 
per  turn,  and  N  =  frequency,  the  E.M.F.  induced  in  the 
machine  is  E  =  4«4>7V10~8  volts,  independent  of  the  num- 
ber of  poles,  of  series  or  multiple  connection  of  the  arma- 
ture, whether  of  the  ring,  drum,  or  other  type. 

In  an  alternator  or  transformer,  if  n  is  the  number  of 
turns  in  series,  $  the  maximum  flux  inclosed  per  turn,  and 
JV  the  frequency,  this  formula  gives, 

£avg  =  4  «  4>  JVW  ~  8  volts. 
Since  the  maximum  E.M.F.  is  given  by,  — 

•^maz.  =  £  ^avg 

we  have 

^"max.  =  27r»<S>7V710-8VOltS. 

And  since  the  effective  E.M.F.  is  given  by,  — 


we  have 

£es.  = 

=  4.44  n  4>^10-  8  volts, 

which  is    the   fundamental   formula  of   alternating-current 
induction  by  sine  waves. 


18  AL  TERN  A  TING-CURRENT  PHENOMENA, 

13.  If,  in  a  circuit  of  n  turns,  the  magnetic  flux,  <t>, 
inclosed  by  the  circuit  is  produced  by  the  current  flowing 
in  the  circuit,  the  ratio  — 

flux  X  number  of  turns  X  10~8 


current    . 

is  called  the  inductance,  L,  of  the  circuit,  in  henrys. 

The  product  of  the  number  of  turns,  n,  into  the  maxi- 
mum flux,  <S>,  produced  by  a  current  of  /  amperes  effective, 
or  /  V2  amperes  maximum,  is  therefore  — 

n®  =Z/V2  108; 
and  consequently  the  effective  E.M.F.  of  self-inductance  is: 


E  =  V2 

='  2  TT  NLI  volts. 

The  product,  x  =  2  vNL,  is  of  the  dimension  of  resistance, 
and  is  called  the  reactance  of  the  circuit  ;  and  the  E.M.F. 
of  self-inductance  of  the  circuit,  or  the  reactance  voltage,  is 

E  =  Ix, 

and  lags  90°  behind  the  current,  since  the  current  is  in 
phase  with  the  magnetic  flux  produced  by  the  current, 
and  the  E.M.F.  lags  90°  behind  the  magnetic  flux.  The 
E.M.F.  lags  90°  behind  the  magnetic  flux,  as  it  is  propor- 
tional to  the  change  in  flux  ;  thus  it  is  zero  when  the  mag- 
netism is  at  its  maximum  value,  and  a  maximum  when  the 
flux  passes  through  zero,  where  it  changes  quickest. 


GRAPHIC  REPRESENTA  TION, 


19 


CHAPTER   IV. 

GRAPHIC    REPRESENTATION. 

14.  While  alternating  waves  can  be,  and  frequently  are, 
represented  graphically  in  rectangular  coordinates,  with  the 
time  as  abscissae,  and  the  instantaneous  values  of  the  wave 
as  ordinates,  the  best  insight  with  regard  to  the  mutual 
relation  of  different  alternate  waves  is  given  by  their  repre- 
sentation in  polar  coordinates,  with  the  time  as  an  angle  or 
the  amplitude,  —  one  complete  period  being  represented  by 
one  revolution,  —  and  the  instantaneous  values  as  radii 
vectores. 


Fig.  8. 


Thus  the  two  waves  of  Figs.  2  and  3  are  represented  in 
polar  coordinates  in  Figs.  8  and  9  as  closed  characteristic 
curves,  which,  by  their  intersection  with  the  radius  vector, 
give  the  instantaneous  value  of  the  wave,  corresponding  to 
the  time  represented  by  the  amplitude  of  the  radius  vector. 

These  instantaneous  values  are  positive  if  in  the  direction 
of  the  radius  vector,  and  negative  if  in  opposition.  Hence 
the  two  half-waves  in  Fig.  2  are  represented  by  the  same 


20 


ALTERNA TING-CURRENT  PHENOMENA. 


polar  characteristic  curve,  which  is  traversed  by  the  point  of 
intersection  of  the  radius  vector  twice  per  period,  —  once 
in  the  direction  of  the  vector,  giving  the  positive  half-wave, 


Fig.  9.  B,  Fig.  10. 

and  once  in  opposition  to  the  vector,  giving  the  negative 
half-wave.  In  Figs.  3  and  9,  where  the  two  half-waves  are 
different,  they  give  different  polar  characteristics. 

15.  The  sine  wave,  Fig.  1,  is  represented  in  polar 
coordinates  by  one  circle,  as  shown  in  Fig.  10.  The 
diameter  of  the  characteristic  curve  of  the  sine  wave, 
1=  OC,  represents  the  intensity  of  the  wave ;  and  the  am- 
plitude of  the  diameter,  OC,  /_&  =  AOC,  is  thefl/iase  of  the 
wave,  which,  therefore,  is  represented  analytically  by  the 

function :  — 

t  =  /cos  (<£  —  w), 

where  </>  =  2  IT  /  /  T is  the  instantaneous  value  of  the  ampli- 
tude corresponding  to  the  instantaneous  value,  2,  of  the  wave. 

The  instantaneous  values  are  cut  out  on  the  movable  ra- 
dius vector  by  its  intersection  with  the  characteristic  circle. 
Thus,  for  instance,  at  the  amplitude  AOBl  =  ^  =  2  ^  /  T 
(Fig.  10),  the  instantaneous  value  is  OB' ;  at  the  amplitude 
AO£2  =  <f>2  =  27T/2/  T,  the  instantaneous  value  is  ~OJ3",  and 
negative,  since  in  opposition  to  the  radius  vector  OBZ. 

The  characteristic  circle  of  the  alternating  sine  wave  is 
determined  by  the  length  of  its  diameter —  the  intensity 
of  the  wave ;  and  by  the  amplitude  of  the  diameter  —  the 
phase  of  the  wave. 


GRAPHIC  REPRESENTATION.  21 

Hence,  wherever  the  integral  value  of  the  wave  is  con- 
sidered alone,  and  not  the  instantaneous  values,  the  charac- 
teristic circle  may  be  omitted  altogether,  and  the  wave 
represented  in  intensity  and  in  phase  by  the  diameter  of 
the  characteristic  circle. 

Thus,  in  polar  coordinates,  the  alternate  wave  is  repre- 
sented in  intensity  and  phase  by  the  length  and  direction  of 
a  vector,  OC,  Fig.  10,  and  its  analytical  expression  would 
then  be  c  =  OC  cos  (<f>  —  w). 

Instead  of  the  maximum  value  of  the  wave,  the  effective 
value,  or  square  root  of  mean  square,  may  be  used  as  the 
vector,  which  is  more  convenient  ;  and  the  maximum  value 
is  then  V2  times  the  vector  OC,  so  that  the  instantaneous 
values,  when  taken  from  the  diagram,  have  to  be  increased 
by  the  factor  V2. 

Thus  the  wave, 

l>  =  £  cos 

=  B  cos  (</>  -  fy 
is  in  Fig.  10#  represented  by 

T) 

vector    OB  =  —  ,    of    phase 

A  OB  =  G!  ;  and  the  wave, 
c=  Ccos 


is  in  Fig.  10#  represented  by  vector  OC=—j=,  of  phase 
AOC=  -£* 

The  former  is  said  to  lag  by  angle  ^,  the  latter  to  lead 
by  angle  £2,  with  regard  to  the  zero  position. 

The  wave  b  lags  by  angle  (o^  +  £2)  behind  wave  c,  or  c 
leads  b  by  angle  (wx  +  £2). 

16.  To  combine  different  sine  waves,  their  graphical  rep- 
resentations, or  vectors,  are  combined  by  the  parallelogram 
law. 

If,  for  instance,  two  sine  waves,  OB  and  OC  (Fig.  11), 
are  superposed,  —  as,  for  instance,  two  E.M.F's.  acting  in 
the  same  circuit,  —  their  resultant  wave  is  represented  by 


22 


ALTERNATING-CURRENT  PHENOMEA?A. 


OD,  the  diagonal  of  a  parallelogram  with  OB  and  OC  as 
sides. 

For  at  any  time,  /,  represented  by  angle  <f>  =  AOX,  the 
instantaneous  values  of  the  three  waves,  OB,  OC,  OD,  are 
their  projections  upon  OX,  and  the  sum  of  the  projections 
of  OB  and  OC  is  equal  to  the  projection  of  OD  ;  that  is,  the 
instantaneous  values  of  the  wave  OD  are  equal  to  the  sum 
of  the  instantaneous  values  of  waves  OB  and  OC. 

From  the  foregoing  considerations  we  have  the  con- 
clusions : 

The  sine  wave  is  represented  graphically  in  polar  coordi- 
nates by  a  vector,  which  by  its  length,  OC,  denotes  the  in- 


Fig.   11. 


tensity,  and  by  its  amplitude,  AOC,  the  phase,  of  the  sine 
wave. 

Sine  waves  are  combined  or  resolved  graphically,  in  polar 
coordinates,  by  the  law  of  parallelogram  or  tJie  polygon  of 
sine  waves. 

Kirchhoff's  laws  now  assume,  for  alternating  sine  waves, 
the  form  :  — 

a.)  The  resultant  of  all  the  E.M.Fs.  in  a  closed  circuit, 
as  found  by  the  parallelogram  of  sine  waves,  is  zero  if 
the  counter  E.M.Fs.  of  resistance  and  of  reactance  are 
included. 

b.}    The  resultant  of  all  the  currents  flowing  towards  a 


GRAPHIC  REPRESENTATION. 


23 


distributing  point,  as  found   by  the  parallelogram  of   sine 
waves,  is  zero. 

The  energy  equation  expressed  graphically  is  as  follows  : 
The  power  of  an  alternating-current  circuit  is  repre- 
sented in  polar  coordinates  by  the  product  of  the  current  , 
/,  into  the  projection  of  the  E.M.F.,  E,  upon  the  current,  or 
by  the  E.M.F.,  E,  into  the  projection  of  the  current,  /,  upon 
the  E.M.F.,  or  by  IE  cos 


17.  Suppose,  as  an  instance,  that  over  a  line  having  the 
resistance,  r,  and  the  reactance,  x  =  ZirNL,  —  where  N  = 
frequency  and  L  =  inductance,  —  a  current  of  /  amperes 
be  sent  into  a  non-inductive  circuit  at  an  E.M.F.  of  E 


Fig.  12. 

volts.     What  will  be  the  E.M.F.  required  at  the  generator 
end  of  the  line  ? 

In  the  polar  diagram,  Fig.  12,  let  the  phase  of  the  cur- 
rent be  assumed  as  the  initial  or  zero  line,  Of.  Since  the 
receiving  circuit  is  non-inductive,  the  current  is  in  phase 
with  its  E.M.F.  Hence  the  E.M.F.,  E,  at  the  end  of  the 
line,  impressed  upon  the  receiving  circuit,  is  represented  by 
a  vector,  OE.  To  overcome  the  resistance,  r,  of  the  line, 
an  E.M.F.,  Ir,  is  required  in  phase  with  the  current,  repre- 
sented by  OEr  in  the  diagram.  The  self-inductance  of  the 
line  induces  an  E.M.F.  which  is  proportional  to  the  current 
/  and  reactance  x,  and  lags  a  quarter  of  a  period,  or  90°, 
behind  the  current.  To  overcome  this  counter  E.M.F. 


24 


ALTERNA TING-CURRENT  PHENOMENA. 


of  self-induction,  an  E.M.F.  of  the  value  Ix  is  required, 
in  phase  90°  ahead  of  the  current,  hence  represented  by 
vector  OEX.  Thus  resistance  consumes  E.M.F.  in  phase, 
and  reactance  an  E.M.F.  90°  ahead  of  the  current.  The 
E.M.F.  of  the  generator,  E0,  has  to  give  the  three  E.M.Fs., 
E,  Ery  and  Ex,  hence  it  is  determined  as  their  resultant. 
Combining  by  the  parallelogram  law,  OEr  and  OEX,  give 
OEZ,  the  E.M.F.  required  to  overcome  the  impedance  of 
the  line,  and  similarly  OEZ  and  OE  give  OE0,  the  E.M.F. 
required  at  the  generator  side  of  the  line,  to  yield  the 
E.M.F.  E  at  the  receiving  end  of  the  line.  Algebraically, 
we  get  from  Fig.  12 — 


or,  E   =  VX2  —  (/*)2  -  Jr. 

In  this  instance  we  have  considered  the  E.M.F.  con- 
sumed by  the  resistance  (in  phase  with  the  current)  and 
the  E.M.F.  consumed  by  the  reactance  (90°  ahead  of  the 
current)  as  parts,  or  components,  of  the  impressed  E.M.F., 
E0,  and  have  derived  E0  by  combining  Er,  Ex,  and  E. 


E'. 


E?    0 


Fig.  13. 


18.  We  may,  however,  introduce  the  effect  of  the  induc- 
tance directly  as  an  E.M.F.,  Ex  ,  the  counter  E.M.F.  of 
self-induction  =  Ix,  and  lagging  90°  behind  the  current ;  and 
the  E.M.F.  consumed  by  the  resistance  as  a  counter  E.M.F., 
Ef  =  Ir,  but  in  opposition  to  the  current,  as  is  done  in  Fig. 
13  ;  and  combine  the  three  E.M.Fs.  E0,  EJ,  Ex ,  to  form  a 
resultant  E.M.F.,  E,  which  is  left  at  the  end  of  the  line- 


GRAPHIC  REPRESENTA  TION. 


25 


Ef  and  £a!  combine  to  form  Eg)  the  counter  E.M.F.  of 
impedance ;  and  since  Eg  and  E0  must  combine  to  form 
E,  E0  is  found  as  the  side  of  a  parallelogram,  OE0EEg) 
whose  other  side,  O£z',  and  diagonal,  OE,  are  given. 

Or  we  may  say  (Fig.  14),  that  to  overcome  the  counter 
E.M.F.  of  impedance,  OEZ,  of  the  line,  the  component,  OEZ, 
of  the  impressed  E.M.F.  is  required  which,  with  the  other 
component  OE,  must  give  the  impressed  E.M.F.,  OE0. 

As  shown,  we  can  represent  the  E.M.Fs.  produced  in  a 
circuit  in  two  ways  —  either  as  counter  E.M.Fs.,  which  com- 
bine with  the  impressed  E.M.F.,  or  as  parts,  or  components, 


E.V  o 


Fig.  14. 

of  the  impressed  E.M.F.,  in  the  latter  case  being  of  opposite 
phase.  According  to  the  nature  of  the  problem,  either  the 
one  or  the  other  way  may  be  preferable. 

As  an  example,  the  E.M.F.  consumed  by  the  resistance 
is  Ir,  and  in  phase  with  the  current  ;  the  counter  E.M.F. 
of  resistance  is  in  opposition  to  the  current.  '  The  E.M.F. 
consumed  by  the  reactance  is  Ix,  and  90°  ahead  of  the  cur- 
rent, while  the  counter  E.M.F.  of  reactance  is  90°  behind 
the  current ;  so  that,  if,  in  Fig.  15,  OI,  is  the  current,  — 

OEr    =  E.M.F.  consumed  by  resistance, 
OEr'  =  counter  E.M.F.  of  resistance, 
OEX    =  E.M.F.  consumed  by  inductance, 
OEX'  =  counter  E.M.F.  of  inductance, 
OEZ    =  E.M.F.  consumed  by  impedance, 
OEt '  =  counter  E.M.F.  of  impedance. 


26  ALTERNATING-CURRENT  PHENOMENA. 

Obviously,  these  counter  E.M.Fs.  are  different  from,  for 
instance,  the  counter  E.M.F.  of  a  synchronous  motor,  in  so 
far  as  they  have  no  independent  existence,  but  exist  only 
through,  and  as  long  as,  the  current  flows.  In  this  respect 
they  are  analogous  to  the  opposing  force  of  friction  in 
mechanics. 

if. 


\f 
—X« 


Fig.   15. 


19.    Coming  back  to  the  equation  found  for  the  E.M.F. 
at  the  generator  end  of  the  line,  — 


we  find,  as  the  drop  of  potential  in  the  line 


A  E  =  E  —  E  =  V£      />'2       /*2  —  E. 


This  is  different  from,  and  less  than,  the  E.M.F.  of 
impedance  — 

Hence  it  is  wrong  to  calculate  the  drop  of  potential  in  a 
circuit  by  multiplying  the  current  by  the  impedance  ;  and  the 
drop  of  potential  in  the  line  depends,  with  a  given  current 
fed  over  the  line  into  a  non-inductive  circuit,  not  only  upon 
the  constants  of  the  line,  r  and  *,  but  also  upon  the  E.M.F., 
E,  at  end  of  line,  as  can  readily  be  seen  from  the  diagrams. 

20.  If  the  receiver  circuit  is  inductive,  that  is,  if  the 
current,  /,  lags  behind  the  E.M.F.,  E,  by  an  angle  w,  and 
we  choose  again  as  the  zero  line,  the  current  OI  (Fig.  16), 
the  E.M.F.,  OE  is  ahead  of  the  current  by  angle  £.  The 


GRAPHIC  REPRESENTA  TION. 


27 


E.M.F.  consumed  by  the  resistance,  Ir,  is  in  phase  with  the 
current,  and  represented  by  OEr;  the  E.M.F.  consumed 
by  the  reactance,  Ix,  is  90°  ahead  of  the  current,  and  re- 
presented by  OEX.  Combining  OE,  OEr,  and  OEX,  we 
get  OE0,  the  E.M.F.  required  at  the  generator  end  of  the 
line.  Comparing  Fig.  16  with  Fig.  13,  we  see  that  in 
the  former  OE0  is  larger  ;  or  conversely,  if  E0  is  the  same, 
E  will  be  less  with  an  inductive  load.  In  other  words, 
the  drop  of  potential  in  an  inductive  line  is  greater,  if  the 
receiving  circuit  is  inductive,  than  if  it  is  non-inductive. 
From  Fig.  16, — 

E0  =  V(^  cos  w  +  Ir)2  -f-  (E  sin  w  +  Ix)z. 


Fig.   18. 

If,  however,  the  current  in  the  receiving  circuit  is 
leading,  as  -is  the  case  when  feeding  condensers  or  syn- 
chronous motors  whose  counter  E.M.F.  is  larger  than  the 
impressed  E.M.F.,  then  the  E.M.F.  will  be  represented,  in 
Fig.  17,  by  a  vector,  OE,  lagging  behind  the  current,  Of, 
by  the  angle  of  lead  £';  and  in  this  case  we  get,  by 
combining  OE  with  OEr,  in  phase  with  the  current,  and 
OEX,  90°  ahead  of  the  current,  the  generator  E.M.F.,  OE~0, 
which  in  this  case  is  not  only  less  than  in  Fig.  16  and  in 
Fig.  13,  but  may  be  even  less  than  E ;  that  is,  the  poten- 
tial rises  in  the  line.  In  other  words,  in  a  circuit  with 
leading  current,  the  self-induction  of  the  line  raises  the 
potential,  so  that  the  drop  of  potential  is  less  than  with 


28 


AL  TERN  A  TING-  CURRENT  PHENOMENA. 


a  non-inductive  load,  or  may  even  be  negative,  and  the 
voltage  at  the  generator  lower  than  at  the  other  end  of 
the  line. 

These  diagrams,  Figs.  13  to  17,  can  be  considered  polar 
diagrams  of  an  alternating-current  generator  of  an  E.M.F., 
E0>  a  resistance  E.M.F.,  Er  =  fr,  a  reactance  E.M.F., 
Ex  =  fx,  and  a  difference  of  potential,  E,  at  the  alternator 
terminals;  and  we  see,  in  this  case,  that  with  an  inductive 
load  the  potential  difference  at  the  alternator  terminals  will 
be  lower  than  with  a  non-inductive  load,  and  that  with  a 
non-inductive  load  it  will  be  lower  than  when  feeding  into 


'E. 


Fig.  17. 

a  circuit  with  leading  current,  as,  for  instance,  a  synchro- 
nous motor  circuit  under  the  circumstances  stated  above. 

21.  As  a  further  example,  we  may  consider  the  dia- 
gram of  an  alternating-current  transformer,  feeding  through 
its  secondary  circuit  an  inductive  load. 

For  simplicity,  we  may  neglect  here  the  magnetic 
hysteresis,  the  effect  of  which  will  be  fully  treated  in  a 
separate  chapter  on  this  subject. 

Let  the  time  be  counted  from  the  moment  when  the 
magnetic  flux  is  zero.  The  phase  of  the  flux,  that  is,  the 
amplitude  of  its  maximum  value,  is  90°  in  this  case,  and, 
consequently,  the  phase  of  the  induced  E.M.F.,  is  180°, 


GRAPHIC  REPRESEiVTA  TIOiV. 


29 


since  the  induced  E.M.F.  lags  90°  behind  the  inducing 
flux.  Thus  the  secondary  induced  E.M.F.,  JE1,  will  be 
represented  by  a  vector,  O£l}  in  Fig.  18,  at  the  phase 
180°.  The  secondary  current,  flf  lags  behind  the  E.M.F., 
Elt  by  an  angle  a>1}  which  is  determined  by  the  resistance 
and  inductance  of  the  secondary  circuit ;  that  is,  by  the 
load  in  the  secondary  circuit,  and  is  represented  in  the  dia- 
gram by  the  vector,  OFl}  of  phase  180  +  Gj. 


Fig.  18. 


Instead  of  the  secondary  current,  flt  we  plot,  however, 


the  secondary  M.M.F., 


where  n1  is  the  number 
This. 


of  secondary  turns,  and  $l  is  given  in  ampere-turns. 
makes  us  independent  of  the  ratio  of  transformation. 

From  the  secondary  induced  E.M.F.,  Ely  we  get  the  flux» 
3>,  required  to  induce  this  E.M.F.,  from  the  equation  — 


where  — 

£i  =  secondary  induced  E.M.F.  ,  in  effective  volts, 
JV  =  frequency,  in  cycles  per  second, 
;/1    =  number  of  secondary  turns, 
3>    =  maximum  value  of  magnetic  flux,  in  webers. 
The  derivation  of   this  equation  has  been   given   in  a 
preceding  chapter. 

This  magnetic  flux,  4>,  is  represented  by  a  vector,  O<b,  at 
the  phase  90°,  and  to  induce  it  an  M.M.F.,  ff  is  required, 


30  ALTERNATING-CURRENT  PHENOMENA. 

which  is  determined  by  the  magnetic  characteristic  of  the 
iron,  and  the  section  and  length  of  the  magnetic  circuit  of 
the  transformer ;  it  is  in  phase  with  the  flux  $,  and  repre- 
sented by  the  vector  OF,  in  effective  ampere-turns. 

The  effect  of  hysteresis,  neglected  at  present,  is  to  shift 
OF  ahead  of  O®,  by  an  angle  a,  the  angle  of  hysteretic 
lead.  (See  Chapter  on  Hysteresis.) 

This  M.M.F.,  O7,  is  the  resultant  of  the  secondary  M.M.F., 
JFlf  and  the  primary  M.M.F.,  SF0;  or  graphically,  OF  is  the 
diagonal  of  a  parallelogram  with  OFl  and  OF0  as  sides.  OF1 
and  OF  being  known,  we  find  OF0,  the  primary  ampere- 
turns,  and  therefrom,  and  the  number  of  primary  turns,  n0, 
the  primary  current,  I0  =  &0/  n0,  which  corresponds  to  the 
secondary  current,  71. 

To  overcome  the  resistance,  r0,  of  the  primary  coil,  an 
E.M.F.,  Er  =  f0r0,  is  required,  in  phase  with  the  current, 
J0,  and  represented  by  the  vector,  OEr. 

To  overcome  the  reactance,  x0  =  2  •*•  n0  L0 ,  of  the  pri- 
mary coil,  an  E.M.F.  Ex  =  I0x0  is  required,  90°  ahead  of 
the  current  f0,  and  represented  by  vector,  OEX. 

The  resultant  magnetic  flux,  4>,  which  in  the  secondary 
coil  induces  the  E.M.F.,  EI}  induces  in  the  primary  coil  an 
E.M.F.  proportional  to  E±  by  the  ratio  of  turns  n0/ nl}  and 
in  phase  with  El ,  or,  — 

77  f         "o   zr 
£,     *m—2£lf 

»1 

•which  is  represented  by  the  vector  OE%'.  To  overcome  this 
counter  E.M.F.,  Et't  a  primary  E.M.F.,  Et,  is  required,  equal 
but  opposite  to  Et',  and  represented  by  the  vector,  OE,. 

The  primary  impressed  E.M.F.,  E0,  must  thus  consist  of 
the  three  components,  OEit  OEr,  and  OEX,  and  is,  there- 
fore, their  resultant  OE0,  while  the  difference  of  phase  in 
the  primary  circuit  is  found  to  be  <30  =  E0OF0. 

22.  Thus,  in  Figs  18  to  20,  the  diagram  of  a  trans- 
former is  drawn  for  the  same  secondary  E.M.F.,  Ev  sec- 


GRAPHIC  REPRESENTA  TION. 


31 


ondary  current,  7L  and  therefore  secondary  M.M.F.,  &v  but 
with  different  conditions  of  secondary  displacement :  — 

In  Fig.  18,  the  secondary  current,  /i ,  lags  60°  behind  the  sec- 
ondary E.M.F.,  EI. 

In  Fig.  19,  the  secondary  current,  71}  is  in  phase  with  the 
secondary  E.M.F.,  El. 

In  Fig.  20,  the  secondary  current,  7: ,  leads  by  60°  the  second- 
ary E.M.F.,  £lf 


These  diagrams  show  that  lag  in  the  secondary  circuit  in- 
creases and  lead  decreases,  the  primary  current  and  primary 
E.M.F.  required  to  produce  in  the  secondary  circuit  the 
same  E.M.F.  and  current ;  or  conversely,  at  a  given  primary 


Fig.  20. 

impressed  E.M.F.,  E0,  the  secondary  E.M.F.,  E^  will  be 
smaller  with  an  inductive,  and  larger  with  a  condenser 
(leading  current)  load,  than  with  a  non-inductive  load. 

At  the   same  time  we   see  that   a  difference  of  phase 
existing  in  the  secondary  circuit  of  a  transformer  reappears 


32  AL  TERNA  TING-CURRENT  PHENOMENA. 

in  the  primary  circuit,  somewhat  decreased  if  leading,  and 
slightly  increased  if  lagging.  Later  we  shall  see  that 
hysteresis  reduces  the  displacement  in  the  primary  circuit, 
so  that,  with  an  excessive  lag  in  the  secondary  circuit,  the 
lag  in  the  primary  circuit  may  be  less  than  in  the  secondary. 
A  conclusion  from  the  foregoing  is  that  the  transformer 
is  not  suitable  for  producing  currents  of  displaced  phase ; 
since  primary  and  secondary  current  are,  except  at  very 
light  loads,  very  nearly  in  phase,  or  rather,  in  opposition, 
to  each  other. 


SYMBOLIC  METHOD. 


CHAPTER    V. 

SYMBOLIC    METHOD. 

23.  The  graphical  method  of  representing  alternating, 
current  phenomena  by  polar  coordinates  of  time  affords  the 
best  means  for  deriving  a  clear  insight  into  the  mutual  rela- 
tion of  the  different  alternating  sine  waves  entering  into  the 
problem.  For  numerical  calculation,  however,  the  graphical 
method  is  generally  not  well  suited,  owing  to  the  widely 
different  magnitudes  of  the  alternating  sine  waves  repre- 
sented in  the  same  diagram,  which  make  an  exact  diagram- 
matic determination  impossible.  For  instance,  in  the  trans- 
former diagrams  (cf.  Figs.  18-20),  the  different  magnitudes 
will  have  numerical  values  in  practice,  somewhat  like  El  — 
100  volts,  and  1-^  =  75  amperes,  for  a  non-inductive  secon- 
dary load,  as  of  incandescent  lamps.  Thus  the  only  reac- 
tance of  the  secondary  circuit  is  that  of  the  secondary  coil, 
or,  x-^  =  .08  ohms,  giving  a  lag  of  ^  =  3.6°.  We  have 
also, 

n^  =      30  turns. 

n0  =    300  turns. 

CFi  =  2250  ampere-turns. 

y    =    100  ampere-turns. 

Er  =      10  volts. 

JSX  =      60  volts. 

E{  =  1000  volts. 

The  corresponding  diagram  is  shown  in  Fig.  21.  Obvi- 
ously, no  exact  numerical  values  can  be  taken  from  a  par- 
allelogram as  flat  as  OF1FF0^  and  from  the  combination  of 
vectors  of  the  relative  magnitudes  1:6: 100. 

Hence  the  importance  of  the  graphical  method  consists 


34 


ALTERNA TING-CURRENT  PHENOMENA. 


not  so  much  in  its  usefulness  for  practical  calculation,  as  to 
aid  in  the  simple  understanding  of  the  phenomena  involved. 

24.  Sometimes  we  can  calculate  the  numerical  values 
trigonometrically  by  means  of  the  diagram.  Usually,  how- 
ever, this  becomes  too  complicated,  as  will  be  seen  by  trying 


Fig.  21. 

to  calculate,  from  the  above  transformer  diagram,  the  ratio 
of  transformation.  The  primary  M.M.F.  is  given  by  the 
equation  :  — 

ffo  =  Vfr2  +  S^2  +  20^  sin  Wi, 

an  expression  not  well  suited  as  a  starting-point  for  further 
calculation. 

A  method  is  therefore  desirable  which  combines  the 
exactness  of  analytical  calculation  with  the  clearness  of 
the  graphical  representation. 


Fig.  22. 

25.  We  have  seen  that  the  alternating  sine  wave  is 
represented  in  intensity,  as  well  as  phase,  by  a  vector,  Of, 
which  is  determined  analytically  by  two  numerical  quanti- 
ties —  the  length,  Of,  or  intensity  ;  and  the  amplitude,  AOf, 
or  phase  <3,  of  the  wave,  /. 

Instead  of  denoting  the  vector  which  represents  the 
sine  wave  in  the  polar  diagram  by  the  polar  coordinates, 


S  YMB  OL1C  ME  T11OD. 


35 


/  and  <3,  we  can  represent  it  by  its  rectangular  coordinates, 
a  and  b  (Fig.  22),  where  — 

a  =  fcos  u>  is  the  horizontal  component, 

b  =  I  sin  co  is  the  vertical  component  of  the  sine  wave. 

This  representation  of  the  sine  wave  by  its  rectangular 
components  is  very  convenient,  in  so  far  as  it  avoids  the 
use  of  trigonometric  functions  in  the  combination  or  reso- 
lution of  sine  waves. 

Since  the  rectangular  components  a  and  b  are  the  hori- 
zontal and  the  vertical  projections  of  the  vector  represent- 
ing the  sine  wave,  and  the  projection  of  the  diagonal  of  a 
parallelogram  is  equal  to  the  sum  of  the  projections  of  its 
sides,  the  combination  of  sine  waves  by  the  parallelogram 


law  is  reduced  to  the  addition,  or  subtraction,  of  their 
rectangular  components.  That  is, 

Sine  waves  are  combined,  or  resolved,  by  adding,  or 
subtracting,  their  rectangular  components. 

For  instance,  if  a  and  b  are  the  rectangular  components 
of  a  sine  wave,  /,  and  a'  and  b'  the  components  of  another 
sine  wave,  /'  (Fig.  23),  their  resultant  sine  wave,  I0,  has  the 
rectangular  components  a0  —  (a  -f-  a!},  and  b0  =  (b  -f-  b'}. 

To  get  from  the  rectangular  components,  a  and  b,  of  a 
sine  wave,  its  intensity,  i,  and  phase,  o>,  we  may  combine  a 
and  b  by  the  parallelogram,  and  derive,  — 


tan 


36  AL  TERN  A  TING-CURRENT  PHENOMENA  . 

Hence  we  can  analytically  operate  with  sine  waves,  as 
with  forces  in  mechanics,  by  resolving  them  into  their 
rectangular  components. 

26.  To  distinguish,  however,  the  horizontal  and  the  ver- 
tical components  of  sine  waves,  so  as  not  to  be  confused  in 
lengthier  calculation,  we  may  mark,  for  instance,  the  vertical 
components,  by  a  distinguishing  index,  or  the  addition  of 
an  otherwise  meaningless  symbol,  as  the  letter  /,  and  thus 
represent  the  sine  wave  by  the  expression,  — 

I=a 


which  now  has  the  meaning,  that  a  is  the  horizontal  and  b 
the  vertical  component  of  the  sine  wave  /;  and  that  both 
components  are  to  be  combined  in  the  resultant  wave  of 
intensity,  —  _ 

/  =  V^  +  //2, 

and  of  phase,  tan  <3  =  b  /  a. 

Similarly,  a  —jb,  means  a  sine  wave  with  a  as  horizon- 
tal, and  —  b  as  vertical,  components,  etc. 

Obviously,  the  plus  sign  in  the  symbol,  a  -f-  jb,  does  not 
imply  simple  addition,  since  it  connects  heterogeneous  quan- 
tities —  horizontal  and  vertical  components  —  but  implies 
combination  by  the  parallelogram  law. 

For  the  present,/  is  nothing  but  a  distinguishing  index, 
and  otherwise  free  for  definition  except  that  it  is  not  an 
.ordinary  number. 

27.  A  wave  of  equal  intensity,  and  differing  in  phase 
from  the  wave  a  +  jb  by  180°,  or  one-half  period,  is  repre- 
sented in  polar  coordinates  by  a  vector  of  opposite  direction, 
and  denoted  by  the  symbolic  expression,  —  a  —  jb.  Or  — 

Multiplying  the  symbolic  expression,  a  +  jb,  of  a  sine  wave 
by  —  1  weans  reversing'  the  wave,  or  rotating  it  through  180°, 
or  one-half  period. 

A  wave  of  equal  intensity,  but  lagging  90°,  or  one- 
quarter  period,  behind  a  -f  jb,  has  (Fig.  24)  the  horizontal 


SYMBOLIC  METHOD.  37 

component,  —  b,  and  the  vertical  component,  a,  and  is  rep- 
resented symbolically  by  the  expression,  ja  —  b, 
Multiplying,  however,  a  +  jb  by/,  we  get  :  — 


therefore,  if  we  define  the  heretofore  meaningless  symbol, 
j,  by  the  condition,  — 

y2  =  -  i, 

we  have  — 

/(*+/*)  =ja  —  1>; 

hence  :  — 

Multiplying  the  symbolic  expression,  a  -\-  jb,  of  a  sine  wave 
by  j  means  rotating  the  wave  through  90°,  or  one-quarter  pe- 
riod ;  tJiat  is,  retarding  the  wave  through  one-quarter  period. 


Fig.  24. 

Similarly,  — 

Multiplying  by  —  j  means  advancing  the  wave  through 
one-quarter  period. 

since  y'2  =  —  1,  j  =  V— 1 ; 

that  is,  — 

j  is  the  imaginary  unit,  and  the  sine  wave  is  represented 
by  a  complex  imaginary  quantity,  a  -+-  jb. 

As  the  imaginary  unit  j  has  no  numerical  meaning  in 
the  system  of  ordinary  numbers,  this  definition  of/  =  V— 1 
does  not  contradict  its  original  introduction  as  a  distinguish- 
ing index.  For  a  more  exact  definition  of  this  complex 
imaginary  quantity,  reference  may  be  made  to  the  text  books 
of  mathematics. 

28.  In  the  polar  diagram  of  time,  the  sine  wave  is 
represented  in  intensity  as  well  as  phase  by  one  complex 
quantity  — 


38  ALTERNATING-CURRENT  PHENOMENA. 

where  a  is  the  horizontal  and  b  the  vertical  component  of 
the  wave  ;  the  intensity  is  given  by  — 

the  phase  by  — 

tan  <o  =  -  , 
a 
and 

a  =  i  cos  to, 

b  =  i  sin  w  ; 

hence  the  wave  a  +jb  can  also  be  expressed  by  — 
/  (cos  <i>  -\-j  sin  <3), 

or,  by  substituting  for  cos  w  and  sin  w  their  exponential 

expressions,  we  obtain  — 

id™. 

Since  we  have  seen  that  sine  waves  may  be  combined 
or  resolved  by  adding  or  subtracting  their  rectangular  com- 
ponents, consequently  :  — 

Sine  waves  may  be  combined  or  resolved  by  adding  or 
subtracting  their  complex  algebraic  expressions. 

For  instance,  the  sine  waves,  — 

a  +jb 
and 


combined  give  the  sine  wave  — 

7-  (a  + 

It  will  thus  be  seen  that  the  combination  of  sine  waves 
is  reduced  to  the  elementary  algebra  of  complex  quantities. 

29.    If  /=  i  +/z'  is  a  sine  wave  of  alternating  current, 
and  r  is  the  resistance,  the  E.M.F.  consumed  by  the  re- 
sistance is  in  phase  with  the  current,  and  equal  to  the  prod- 
uct of  the  current  and  resistance.     Or  — 
rl  '  —  ri  -\-  jri'  . 

If  L  is  the  inductance,  and  x  =  2  TT  NL  the  reactance, 
the    E.M.F.   produced    by  the    reactance,    or   the   counter 


SYMBOLIC  METHOD.  39 

E.M.F.  of  self-induction,  is  the  product  of  the  current 
and  reactance,  and  lags  90°  behind  the  current  ;  it  is, 
therefore,  represented  by  the  expression  — 


The  E.M.F.  required  to  overcome  the  reactance  is  con-  , 
sequently  90°  ahead  of  the  current  (or,  as  usually  expressed,-** 
the  current  lags  90°  behind  the  E.M.F.),  and  represented 
by  the  expression  — 

— jxl =  — jxi  -f-  xi'. 

Hence,  the  E.M.F.  required  to  overcome  the  resistance, 
r,  and  the  reactance,  x,  is  — 


that  is  — 

Z  =  r  —  jx  is  the  expression  of  the  impedance  of  the  cir- 
cuit, in  complex  quantities. 

Hence,  if  /  =  i  -\-ji'  is  the  current,  the  E.M.F.  required 
to  overcome  the  impedance,  Z  =  r  —  jx,  is  — 


hence,  sincey"2  =  —  1 


or,  if  E  =  e  -\-  je'  is  the  impressed  E.M.F.,  and  Z  =  r  —  jx 
the  impedance,  the  current  flowing  through  the  circuit  is  :  — 


or,  multiplying  numerator  and  denominator  by  (r+jx)  to 
eliminate  the  imaginary  from  the  denominator,  we  have  — 


T  _ 


or,  if  E  =  e  -\-je'  is  the  impressed  E.M.F.,  and  7  =  i  '  -\-  ji' 
the  current  flowing  in  the  circuit,  its  impedance  is  — 

0  +./>')  O'-./*'')      «'+^*''  .       '  ~  ei' 

' 


40  ALTERNATING-CURRENT  PHENOMENA. 

30.    If  C  is  the  capacity  of   a  condenser  in    series    in 
a  circuit  of  current  I  =  i  +  //',  the  E.M.F.  impressed  upon 

the  terminals  of  the  condenser  is  E  =  -  -  ,  90°  behind 
the  current  ;  and  may  be  represented  by  —  -  -  ,  or  jx^  /, 

where  x^  =  -  is  the  capacity  reactance  or  condensatice 
2  TT  NC 

of  the  condenser. 

Capacity  reactance  is  of  opposite  sign  to  magnetic  re- 
actance ;  both  may  be  combined  in  the  name  reactance. 

We  therefore  have  the  conclusion  that 

If  r  =  resistance  and  L  =  inductance, 

then  x  =  2  IT  NL  =  magnetic  reactance. 

If  C  =  capacity,  x^  =  -  =  capacity  reactance,  or  conden- 
sance  ; 

Z  =  r  —  j  (x  —  JCi),  is  the  impedance  of  the  circuit 
Ohm's  law  is  then  reestablished  as  follows  : 


,  -,  . 

The  more  general  form  gives  not  only  the  intensity  of 
the  wave,  but  also  its  phase,  as  expressed  in  complex 
quantities. 

31.  Since  the  combination  of  sine  waves  takes  place  by 
the  addition  of  their  symbolic  expressions,  Kirchhoff's  laws 
are  now  reestablished  in  their  original  form  :  — 

a.}  The  sum  of  all  the  E.M.Fs.  acting  in  a  closed  cir- 
cuit equals  zero,  if  they  are  expressed  by  complex  quanti- 
ties, and  if  the  resistance  and  reactance  E.M.Fs.  are  also 
considered  as  counter  E.M.Fs. 

b.)  The  sum  of  all  the  currents  flowing  towards  a  dis- 
tributing point  is  zero,  if  the  currents  are  expressed  as 
complex  quantities. 


SYMBOLIC  METHOD.  41 

If  a  complex  quantity  equals  zero,  the  real  part  as  well 
as  the  imaginary  part  must  be  zero  individually,  thus  if 
a  +jb  =  0,  a  =  0,  b  =  0. 

Resolving  the  E.M.Fs.  and  currents  in  the  expression  of 
Kirchhoff 's  law,  we  find  :  — 

a.}  The  sum  of  the  components,  in  any  direction,  of  all 
the  E.M.Fs.  in  a  closed  circuit,  equals  zero,  'if  the  resis- 
tance and  reactance  are  considered  as  counter  E.M.Fs. 

b.}  The  sum  of  the  components,  in  any  direction,  of  all 
the  currents  flowing  to  a  distributing  point,  equals  zero. 

Joule's  Law  and  the  energy  equation  do  not  give  a 
simple  expression  in  complex  quantities,  since  the  effect  or 
power  is  a  quantity  of  double  the  frequency  of  the  current 
or  E.M.F.  wave,  and  therefore  requires  for  its  representa- 
tion as  a  vector,  a  transition  from  single  to  double  fre- 
quency, as  will  be  shown  in  chapter  XII. 

In  what  follows,  complex  vector  quantities  will  always 
be  denoted  by  dotted  capitals  when  not  written  out  in  full ; 
absolute  quantities  and  real  quantities  by  undotted  letters. 

32.  Referring  to  the  instance  given  in  the  fourth 
chapter,  of  a  circuit  supplied  with  an  E.M.F.,  E,  and  a  cur- 
rent, 7,  over  an  inductive  line,  we  can  now  represent  the 
impedance  of  the  line  by  Z  =  r  —  jx,  where  r  =  resistance, 
x  =  reactance  of  the  line,  and  have  thus  as  the  E.M.F. 
at  the  beginning  of  the  line,  or  at  the  generator,  the 

expression  — 

E0  =  E  +  ZI. 

Assuming  now  again  the  current  as  the  zero  line,  that 
is,  /  =  /,  we  have  in  general  — 

E0  =  E  -f  ir  —jix ; 

hence,  with  non-inductive  load,  or  E  =  e, 
E0=(e  +  ir)  -jix, 


+  /r)2  +  (/X)2,     tan  S>0  = 


42  ALTERNATING-CURRENT  PHENOMENA. 

In  a    circuit  with    lagging  current,  that    is,  with    leading 
E.M.F.,  E  =  e  -je',  and 


*-*)2>     tan  <S0 


e  +  /> 

In  a    circuit  with  leading    current,  that    is,    with    lagging 
E.M.F.,  E  =  *  +>',  and 


—  /V)  ,     tan  w0  = 
values  which  easily  permit  calculation. 


TOPOGRAPHIC  METHOD.  43 


CHAPTER    VI. 

TOPOGRAPHIC    METHOD. 

33.  In  the  representation  of  alternating  sine  waves  by 
vectors  in  a  polar  diagram,  a  certain  ambiguity  exists,  in  so 
far  as  one  and  the  same  quantity  —  an  E.M.F.,  for  in- 
stance —  can  be  represented  by  two  vectors  of  opposite 
direction,  according  as  to  whether  the  E.M.F.  is  considered 
as  a  part  of  the  impressed  E.M.F.,  or  as  a  counter  E.M.F. 
This  is  analogous  to  the  distinction  between  action  and 
reaction  in  mechanics. 


Further,  it  is  obvious  that  if  in  the  circuit  of  a  gener- 
ator, G  (Fig.  25),  the  current  flowing  from  terminal  A  over 
resistance  R  to  terminal  B,  is  represented  by  a  vector  OI 
(Fig.  26),  or  by  /=  i  -\-ji',  the  same  current  can  be  con- 
sidered as  flowing  in  the  opposite  direction,  from  terminal 
B  to  terminal  A  in  opposite  phase,  and  therefore  represented 
by  a  vector  OI-±  (Fig.  26),  or  by  7l  =  —  i  —ji'> 

Or,  if  the  difference  of  potential  from  terminal  B  to 
terminal  A  is  denoted  by  the  E  =  e  +  je' ,  the  difference 
of  potential  from  A  to  B  is  El  =  —  e  —  je' . 


44 


ALTERNA TING-CURRENT  PHENOMENA. 


Hence,  in  dealing  with  alternating-current  sine  waves, 
it  is  necessary  to  consider  them  in  their  proper  direction 
with  regard  to  the  circuit.  Especially  in  more  complicated 
circuits,  as  interlinked  polyphase  systems,  careful  attention 
has  to  be  paid  to  this  point. 


-*' 

Fig.  28. 

34.  Let,  for  instance,  in  Fig.  27,  an  interlinked  three- 
phase  system  be  represented  diagrammatically,  as  consist- 
ing of  three  E.M.Fs.,  of  equal  intensity,  differing  in  phase 
by  one-third  of  a  period.  Let  the  E.M.Fs.  in  the  direction 


Fig.  27 


from  the  common  connection  O  of  the  three  branch  circuits 
to  the  terminals  A19  A2,AB,  be  represented  by  Elt  E2,  £3. 
Then  the  difference  of  potential  from  A2  to  A±  is  £z  —  £lf 


since  the  two  E.M.Fs.,  El  and 


are  connected  in  cir- 


cuit between  the  terminals  A,   and  A*,  in  the  direction, 


TOPOGRAPHIC  METHOD.  45 

Al  —  O  —  A2;  that  is,  the  one,  Ez,  in  the  direction  OA2, 
from  the  common  connection  to  terminal,  the  other,  JS1,  in 
the  opposite  direction,  A^O,  from  the  terminal  to  common 
connection,  and  represented  by  —  El.  Conversely,  the  dif- 
ference of  potential  from  A1  to  Az  is  El  —  Ez. 

It  is  then  convenient  to  go  still  a  step  farther,  and 
drop,  in  the  diagrammatic  representation,  the  vector  line 
altogether ;  that  is,  denote  the  sine  wave  by  a  point  only,, 
the  end  of  the  corresponding  vector. 

"  Looking  at  this  from  a  different  point  of  view,  it  means 
that  we  choose  one  point  of  the  system  —  for  instance,  the 
common  connection  O  —  as  a  zero  point,  or  point  of  zero 
potential,  and  represent  the  potentials  of  all  the  other  points 
of  the  circuit  by  points  in  the  diagram,  such  that  their  dis- 
tances from  the  zero  point  gives  the  intensity ;  their  ampli- 
tude the  phase  of  the  difference  of  potential  of  the  respective 
point  with  regard  to  the  zero  point ;  and  their  distance  and 
amplitude  with  regard  to  other  points  of  the  diagram,  their 
difference  of  potential  from  these  points  in  intensity  and 
phase. 


Fig.  28. 

Thus,  for  example,  in  an  interlinked  three-phase  system 
with  three  E.M.Fs.  of  equal  intensity,  and  differing  in  phase 
by  one-third  of  a  period,  we  may  choose  the  common  con- 
nection of  the  star-connected  generator  as  the  zero  point, 
and  represent,  in  Fig.  28,  one  of  the  E.M.Fs.,  or  the  poten- 


46 


AL  TERN  A  TING-CURRENT  PHENOMEMA. 


tial  at  one  of  the  three-phase  terminals,  by  point  Er  The 
potentials  at  the  two  other  terminals  will  then  be  given  by 
the  points  Ez  and  E&  which  have  the  same  distance  from 
O  as  Ev  and  are  equidistant  from  E±  and  from  each  other. 
The  difference  of  potential  between  any  pair  of  termi- 
nals —  for  instance  E^  and  E2  —  is  then  the  distance  EZEV 
or  E±EV  according  to  the  direction  considered. 

35.  If  now  the  three  branches  OEV  ~OEZ  and  "OEW  of 
the  three-phase  system  are  loaded  equally  by  three  currents 
equal  in  intensity  and  in  difference  of  phase  against  their 


THUEE-PHA8E  8V8TEM 
48° LAO 


BALANCED  THREE-PHASE  SYSTEM 

NON-INDUCTIVE  LOAD 
E° 


Fig.  29. 


E.M.Fs.,  these  currents  are  represented  in  Fig.  29  by  the 
vectors  07^  =  072  =  Ofs  =  I,  lagging  behind  the  E.M.Fs. 
by  angles  E.O^  =  EZOIZ  =  EZOI&  =  Q. 

Let  the  three-phase  circuit  be  supplied  over  a  line  of 
impedance  Z±  =  r^  —jx\  from  a  generator  of  internal  im- 
pedance Z0  =  x0  -jx0. 

In  phase  OEV  the  E.M.F.  consumed  by  resistance  r^  is 
represented  by  the  distance  E^EJ  =  Irv  in  phase,  that  is 
parallel  with  current  OIV  The  E.M.F.  consumed  by  re- 
actance #!  is  represented  by  E^Ej'  =  Ixv  90°  ahead  of  cur- 


TOPOGRAPHIC  METHOD. 


47 


rent  OIr  The  same  applies  to  the  other  two  phases,  and 
it  thus  follows  that  to  produce  the  E.M.F.  triangle  E^E^E^ 
at  the  terminals  of  the  consumer's  circuit,  the  E.M.F.  tri- 
angle E^E^E?  is  required  at  the  generator  terminals. 

Repeating  the  same  operation  for  the  internal  impedance 
of  the  generator  we  get  E"E'"  =  Iroi  and  parallel  to  OIV 
E'"E°  =  Ixoy  and  90°  ahead  of  ~OTV  and  thus  as  triangle  of 
(nominal)  induced  E.M.Fs.  of  the  generator  E°E£E°. 

In  Fig.  29,  the  diagram  is  shown  for  45°  lag,  in  Fig.  30 
for  noninductive  load,  and  in  Fig.  31  for  45°  lead  of  the 
currents  with  regard  to  their  E.M.Fs. 


BALANCED  THREE 
-PHASE  SYSTEM 

45° LEAD 


THREE-PHASE  CIRCUIT 

80°LA» 

TRANSMISSION  LINE' 

WITH  DISTRIBUTED 

CAPACITY,    INDUCTANCB 

RESISTANCE  AUD  LEAKAQB 

•I, 


Fig.  31. 


Fig.  32. 


As  seen,  the  induced  generator  E.M.F.  and  thus  the 
generator  excitation  with  lagging  current  must  be  higher, 
with  leading  current  lower,  than  at  non-inductive  load,  or 
conversely  with  the  same  generator  excitation,  that  is  the 
same  induced  generator  E.M.F.  triangle  E°E£E°,  the 
E.M.Fs.  at  the  receiver's  circuit,  Ev  Ez,  E9  fall  off  more 
with  lagging,  less  with  leading  current,  than  with  non- 
inductive  load. 

36.  As  further  instance  may  be  considered  the  case  of 
a  single  phase  alternating  current  circuit  supplied  over  a 
cable  containing  resistance  and  distributed  capacity. 


48  ALTERNATING-CURRENT  PHENOMENA. 

Let  in  Fig.  33  the  potential  midway  between  the  two 
terminals  be  assumed  as  zero  point  0.  The  two  terminal 
voltages  at  the  receiver  circuit  are  then  represented  by  the 
points  E  and  El  equidistant  from  0  and  opposite  each  other, 
and  the  two  currents  issuing  from  the  terminals  are  rep- 
resented by  the  points  /  and  I1,  equidistant  from  0  and 
opposite  each  other,  and  under  angle  &  with  E  and  El 
respectively. 

Considering  first  an  element  of  the  line  or  cable  next  to 
the  receiver  circuit.  In  this  an  E.M.F.  EEl  is  consumed 
by  the  resistance  of  the  line  element,  in  phase  with  the 
current  OI,  and  proportional  thereto,  and  a  current  //x  con- 
sumed by  the  capacity,  as  charging  current  of  the  line 
element,  90°  ahead  in  phase  of  the  E.M.F.  OE  and  propor- 
tional thereto,  so  that  at  the  generator  end  of  this  cable 
element  current  and  E.M.F.  are  OI^  and  OEl  respectively. 

Passing  now  to  the  next  cable  element  we  have  again  an 
E.M.F.  E1EZ  proportional  to  and  in  phase  with  the  current 
OI^  and  a  current  IJZ  proportional  to  and  90°  ahead  of  the 
E.M.F.  OEV  and  thus  passing  from  element  to  element 
along  the  cable  to  the  generator,  we  get  curves  of  E.M.Fs. 
e  and  e1,  and  curves  of  currents  i  and  il,  which  can  be  called 
the  topographical  circuit  characteristics,  and  which  corre- 
spond to  each  other,  point  for  point,  until  the  generator 
terminal  voltages  OE0  and  OE0l  and  the  generator  currents 
OI0  and  OIJ  are  reached. 

Again,  adding  'E~Er' =  I0r0  and  parallel  OI0  and  E"E°  = 
I0x0  and  90°  ahead  of  ~OIM  gives  the  (nominal)  induced 
E.M.F.  of  the  generator  OE°,  where  Z0  =  r0  —  jx0  =  inter- 
nal impedance  of  the  generator. 

In  Fig.  33  is  shown  the  circuit  characteristics  for  60° 
lag,  of  a  cable  containing  only  resistance  and  capacity. 

Obviously  by  graphical  construction  the  circuit  character- 
istics appear  more  or  less  as  broken  lines,  due  to  the  neces- 
sity of  using  finite  line  elements,  while  in  reality  when 
calculated  by  the  differential  method  they  are  smooth  curves. 


TOPOGRAPHIC  METHOD. 


49 


37.  As  further  instance  may  be  considered  a  three-phase 
circuit  supplied  over  a  long  distance  transmission  line  of 
distributed  capacity,  self-induction,  resistance,  and  leakage. 

Let,  in  Fig.  38,  O£v  ~OEy  ~OEZ  =  three-phase  E.M.Fs. 
at  receiver  circuit,  equidistant  from  each  other  and  =  E. 

Let  OIV  Oly  Of3  =  three-phase  currents  in  the  receiver 
circuit  equidistant  from  each  other  and  =  /,  and  making 
with  E  the  phase  angle  <3. 

Considering  again  as  in  §  35  the  transmission  line  ele- 
ment by  element,  we  have  in  every  element  an  E.M.F. 
consumed  by  the  resistance  in  phase  with  the  current 
n^  proportional  thereto,  and  an  E.M.F.  E^,  Ef  con- 


sumed by  the  reactance  of  the  line  element,  90°  ahead  of 
the  current  OIV  and  proportional  thereto. 

In  the  same  line  element  we  have  a  current  IJ^  in  phase 
with  the  E.M.F.  OEV  and  proportional  thereto,  representing 
the  loss  of  energy  current  by  leakage,  dielectric  hysteresis, 
etc.,  and  a  current  ^V/',  90°  ahead  of  the  E.M.F.  OEV  and 
proportional  thereto,  the  charging  current  of  the  line  ele- 
ment as  condenser,  and  in  this  manner  passing  along  the 
line,  element  by  element,  we  ultimately  reach  the  generator 
terminal  voltages  E°,  E°,  Es°,  and  generator  currents  //, 
/2°,  78°,  over  the  topographical  characteristics  of  E.M.F.  ev 
ev  es,  and  of  current  iv  z'2,  z'3,  as  shown  in  Fig.  33. 

The  circuit  characteristics  of  current  i  and  of  E.M.F.  e 


50 


ALTERNATING-CURRENT  PHENOMENA. 


correspond  to  each  other,  point  for  point,  the  one  giving  the 
current  and  the  other  the  E.M.F.  in  the  line  element. 


TRANSMISSION 

WITH  DISTRIBUTED 

CAPACITY,    INDUCTANCE 

RESISTANCE  AND  LEAKAGE 
90°  LAO 


Fig.  34. 

Only  the  circuit  characteristics  of  the  first  phase  are 
shown  as  ^  and  z'r  As  seen,  passing  from  the  receiving 
end  towards  the  generator  end  of  the  line,  potential  and 


TRANSMISSION  LINE 

WITH  DISTRIBUTED  CAPACITY,   INDUCTANCE 
RESISTANCE  AND  LEAKAGE 


Fig.  35. 


current  alternately  rise  and  fall,  while  their  phase  angle 
changes  periodically  between  lag  and  lead. 


TOPOGRAPHIC  METHOD.  51 

37.  a.  More  markedly  this  is  shown  in  Fig.  34,  the  topo- 
graphic circuit  characteristic  of  one  of  the  lines  with  90° 
lag  in  the  receiver  circuit.  Corresponding  points  of  the 
two  characteristics  e  and  i  are  marked  by  corresponding 
figures  0  to  16,  representing  equidistant  points  of  the  line. 
The  values  of  E.M.F.,  current  and  their  difference  of  phase 
are  plotted  in  Fig.  35  in  rectangular  co-ordinates  with  the 
distance  as  abscissae,  counting  from  the  receiving  circuit 
towards  the  generator.  As  seen  from  Fig.  35,  E.M.F.  and 
current  periodically  but  alternately  rise  and  fall,  a  maximum 
of  one  approximately  coinciding  with  a  minimum  of  the 
other  and  with  a  point  of  zero  phase  displacement. 

The  phase  angle  between  current  and  E.M.F.  changes 
from  90°  lag  to  72°  lead,  44°  lag,  34°  lead,  etc.,  gradually 
decreasing  in  the  amplitude  of  its  variation. 


52  ALTERNATING-CURRENT  PHENOMENA. 


CHAPTER   VII. 

ADMITTANCE,    CONDUCTANCE,    SUSCEPTANCE. 

38.  If  in  a  continuous-current  circuit,  a  number  of 
resistances,  ?\,  r%,  r3,  .  .  .  are  connected  in  series,  their 
joint  resistance,  R,  is  the  sum  of  the  individual  resistances 

If,  however,  a  number  of  resistances  are  connected  in 
multiple  or  in  parallel,  their  joint  resistance,  R,  cannot 
be  expressed  in  a  simple  form,  but  is  represented  by  the 
expression :  — 

=  J_  _l_  JL  +  J_  + 

/*!  /*2  ^3 

Hence,  in  the  latter  case  it  is  preferable  to  introduce,  in- 
stead of  the  term  resistance,  its  reciprocal,  or  inverse  value, 
the  term  conductance,  g  =  1  /  r.  If,  then,  a  number  of  con- 
ductances, g^,  g^,  gz,  .  .  .  are  connected  in  parallel,  their 
joint  conductance  is  the  sum  of  the  individual  conductances, 
or  G  =  gl  +  gz  +  gz  +  .  .  .  When  using  the  term  con- 
ductance, the  joint  conductance  of  a  number  of  series- 
connected  conductances  becomes  similarly  a  complicated 
expression  — 


Hence  the  term  resistance  is  preferable  in  case  of  series 
connection,  and  the  use  of  the  reciprocal  term  conductance 
in  parallel  connections  ;  therefore, 

The  joint  resistance  of  a  number  of  series-connected  resis- 
tances is  equal  to  the  sum  of  the  individual  resistances  ;  the 


ADMITTANCE,  CONDUCTANCE,  SUSCEPTANCE.         53 

joint  conductance  of  a  number  of  parallel-connected  conduc~ 
tances  is  equal  to  the  sum  of  the  individual  conductances. 

39.  In  alternating-current  circuits,  instead  of  the  term 
resistance  we  have  the  term  impedance,  Z  =  r  —Jx,  with  its 
two  components,  the  resistance,  r,  and  the  reactance,  x,  in  the 
formula  of  Ohm's  law,  E  =  IZ.  The  resistance,  r,  gives 
the  component  of  E.M.F.  in  phase  with  the  current,  or  the 
energy  component  of  the  E.M.F.,  Ir;  the  reactance,  x, 
gives  the  component  of  the  E.M.F.  in  quadrature  with  the 
current,  or  the  wattless  component  of  E.M.F.,  Ix ;  both 
combined  give  the  total  E.M.F., — 


Since  E.M.Fs.  are  combined  by  adding  their  complex  ex- 
pressions, we  have  : 

The  joint  impedance  of  a  number  of  series-connected  impe- 
dances is  the  sum  of  the  individual  impedances,  when  expressed 
in  complex  quantities. 

In  graphical  representation  impedances  have  not  to  be 
added,  but  are  combined  in  their  proper  phase  by  the  law 
of  parallelogram  in  the  same  manner  as  the  E.M.Fs.  corre- 
sponding to  them. 

The  term  impedance  becomes  inconvenient,  however, 
when  dealing  with  parallel-connected  circuits  ;  or,  in  other 
words,  when  several  currents  are  produced  by  the  same 
E.M.F.,  such  as  in  cases  where  Ohm's  law  is  expressed  in 
the  form, 

-I- 

It  is  preferable,  then,  to  introduce  the  reciprocal  of 
impedance,  which  may  be  called  the  admittance  of  the 
circuit,  or 

>-*• 

As  the  reciprocal  of  the  complex  quantity,  Z  =  r  —jx,  the 
admittance  is  a  complex  quantity  also,  or  Y  =  g+jb; 


54  ALTERNATING-CURRENT  PHENOMENA. 

it  consists  of  the  component  g,  which  represents  the  co- 
efficient of  current  in  phase  with  the  E.M.F.,  or  energy 
current,  gEt  in  the  equation  of  Ohm's  law,  — 


and  the  component  b,  which  represents  the  coefficient  of 
current  in  quadrature  with  the  E.M.F.,  or  wattless  com- 
ponent of  current,  bE. 

g  is  called  the  conductance,  and  b  the  susceptance,  of 
the  circuit.  Hence  the  conductance,  g,  is  the  energy  com- 
ponent, and  the  susceptance,  b,  the  wattless  component, 
of  the  admittance,  Y  =  g  -f  jb,  while  the  numerical  value  of 

admittance  is  — 

y  =  Vr1  +  P  ; 

the  resistance,  r,  is  the  energy  component,  and  the  reactance, 
x,  the  wattless  component,  of  the  impedance,  Z  —  r  —  jx, 
the  numerical  value  of  impedance  being  — 

z  =  VV'  +  x\ 

40.  As  shown,  the  term  admittance  implies  resolving 
the  current  into  two  components,  in  phase  and  in  quadra- 
ture with  the  E.M.F.,  or  the  energy  current  and  the  watt- 
less current  ;  while  the  term  impedance  implies  resolving 
the  E.M.F.  into  two  components,  in  phase  and  in  quad- 
rature with  the  current,  or  the  energy  E.M.F.  and  the 
wattless  E.M.F. 

It  must  be  understood,  however,  that  the  conductance 
is  not  the  reciprocal  of  the  resistance,  but  depends  upon 
the  resistance  as  well  as  upon  the  reactance.  Only  when  the 
reactance  x  =  0,  or  in  continuous-current  circuits,  is  the 
conductance  the  reciprocal  of  resistance. 

Again,  only  in  circuits  with  zero  resistance  (r  =  0)  is 
the  susceptance  the  reciprocal  of  reactance  ;  otherwise,  the 
susceptance  depends  upon  reactance  and  upon  resistance. 

The  conductance  is  zero  for  two  values  of  the  resistance  :  — 

1.)  If  r  =  QO  ,  or  x  =  oo  ,  since  in  this  case  no  current 
passes,  and  either  component  of  the  current  =  0. 


ADMITTANCE,  CONDUCTANCE,  SUSCEPTANCE.         55 

2.)  If  r  =  0,  since  in  this  case  the  current  which  passes 
through  the  circuit  is  in  quadrature  with  the  E.M.F.,  and 
thus  has  no  energy  component. 

Similarly,  the  susceptance,  b,  is  zero  for  two  values  of 
the  reactance  :  — 

1.)    If  x  =  oo  ,  or  r  =  oo  . 

2.)    If  *  =  0. 

From  the  definition  of  admittance,  Y '  =  g  +  jbt  as  the 
reciprocal  of  the  impedance,  Z  =  r  —  jxy 

we  have  Y  —  —  ,  or,  g  -f-  jb  = 


Z  r  —jx 

or,  multiplying  numerator  and  denominator  on  the  right  side 

by(r 


hence,  since 

(r-jx)  (r  +»  =  r2  +  x*  =  z\ 


x  r    .    .  x 


, 
and  conversely 


By  these  equations,  the  conductance  and  susceptance  can 
be  calculated  from  resistance  and  reactance,  and  conversely. 
•     Multiplying  the  equations  for^-  and  r,  we  get :  — 

gr     = 

hence, 

an  j  _  1  1  )  the  absolute  value  of 


y       V^"2  +  b* '     )  impedance  ; 
1  1  )  the  absolute  value  of 

admittance. 


56 


AL  TERNA  TING-CURRENT  PHENOMENA. 


41.  If,  in  a  circuit,  the  reactance,  *-,  is  constant,  and  the 
resistance,  r,  is  varied  from  r  =  0  to  r  =  oo  ,  the  susceptance, 
b,  decreases  from  b  =  1  /  x  at  r  =  0,  to  #  =  0  at  r  =  cc ; 
while  the  conductance,  g  —  0  at  r  =  0,  increases,  reaches 
a  maximum  for  r  =  x,  where  g  —  1  /  2  r  is  equal  to  the 
susceptance,  or  g  =  b,  and  then  decreases  again,  reaching 
g  =  0  at  r  =  oo  . 


s 

^N 

V 

\ 

RE; 

CT 

NC 

CO 

NST 

ANT 

-.1 

OH 

MS 

/ 

> 

\ 

s 

\ 

\ 

s 

\ 

s 

\ 

/ 

\ 

x 
/ 

\ 

1 

s 

/ 
'r'  • 

\ 

-/^ 

X 

^ 

fj" 

\ 

1 

\ 

'$ 

* 

\ 

i>S 

S  ^ 

X 

f 

V 

~\ 

^»' 

^^  s 

\ 

/ 

\ 

/ 

\ 

\ 

\ 

/ 

X 

i 

\ 

<^ 

*+*. 

/ 

- 

s' 

j°« 

X 

^^ 

^C: 

~^-^ 

•^ 

.2 

~- 

'V4 

^S 

"~~- 

^ 

^ 

I 

^s 

\ 

<*• 

<» 

^ 

i 

^ 

<. 

I 

-  —  , 

—  -  — 

q 

R 

SIS 

FAN 

OE: 

•,o 

MS 

l.S 

In  Fig.  36,  for  constant  reactance  ^-  =  .5  ohm,  the  vari- 
ation  of  the  conductance,  g,  and  of  the  susceptance,  b,  are 
shown  as  functions  of  the  varying  resistance,  r.  As  shown, 
the  absolute  value  of  admittance,  susceptance,  and  conduc- 
tance are  plotted  in  full  lines,  and  in  dotted  line  the  abso- 
lute value  of  impedance, 


ADMITTANCE,  CONDUCTANCE,  SUSCEPTANCE.         57 

Obviously,  if  the  resistance,  r,  is  constant,  and  the  reac- 
tance, x,  is  varied,  the  values  of  conductance  and  susceptance 
are  merely  exchanged,  the  conductance  decreasing  steadily 
from  g  =  1  /  r  to  0,  and  the  susceptance  passing  from  0  at 
x  =  0  to  the  maximum,  b  =  1  /  2  r  =  g  =1  / '2  x  at  x  =  r, 
and  to  b  =  0  at  x  =  GO  . 

The  resistance,  r,  and  the  reactance,  x,  vary  as  functions 
of  the  conductance,  g,  and  the  susceptance,  b,  in  the  same 
manner  as  g  and  b  vary  as  functions  of  r  and  x. 

The  sign  in  the  complex  expression  of  admittance  is 
always  opposite  to  that  of  impedance ;  this  is  obvious,  since 
if  the  current  lags  behind  the  E.M.F.,  the  E.M.F.  leads  the 
current,  and  conversely. 

We  can  thus  express  Ohm's  law  in  the  two  forms  — 

E  =  IZ, 
I  =£Y, 

and  therefore  — 

The  joint  impedance  of  a  number  of  series-connected  im- 
pedances is  equal  to  the  sum.  of  the  individual  impedances  ; 
the  joint  admittance  of  a  number  of  parallel-connected  admit- 
tances, if  expressed  in  complex  quantities,  is  equal  to  the  sum 
of  the  individual  admittances.  In  diagrammatic  represen- 
tation, combination  by  the  parallelogram  law  takes  the  place 
of  addition  of  the  complex  quantities. 


58  ALTERNATING-CURRENT  PHENOMENA. 


CHAPTER   VIII. 

CIRCUITS    CONTAINING    RESISTANCE,    INDUCTANCE,    AND 
CAPACITY. 

42.  Having,  in  the  foregoing,  reestablished  Ohm's  law 
and  Kirchhoff's  laws  as  being  also  the  fundamental  laws 
of  alternating-current  circuits,  when  expressed  in  their  com- 
plex form, 

E  =  ZS,  or,  /  =  YE, 

and  *%E  =  0  in  a  closed  circuit, 

S/  =  0  at  a  distributing  point, 

where  E,  I,  Z,  Y,  are  the  expressions  of  E.M.F.,  current, 
impedance,  and  admittance  in  complex  quantities,  —  these 
values  representing  not  only  the  intensity,  but  also  the  phase, 
of  the  alternating  wave,  —  we  can  now  —  by  application  of 
these  laws,  and  in  the  same  manner  as  with  continuous- 
current  circuits,  keeping  in  mind,  however,  that  E,  I,  Z,  Y, 
are  complex  quantities  —  calculate  alternating-current  cir- 
cuits and  networks  of  circuits  containing  resistance,  induc- 
tance, and  capacity  in  any  combination,  without  meeting 
with  greater  difficulties  than  when  dealing  with  continuous- 
current  circuits. 

It  is  obviously  not  possible  to  discuss  with  any  com- 
pleteness all  the  infinite  varieties  of  combinations  of  resis- 
tance, inductance,  and  capacity  which  can  be  imagined,  and 
which  may  exist,  in  a  system  or  network  of  circuits  ;  there- 
fore only  some  of  the  more  common  or  more .  interesting 
combinations  will  here  be  considered. 

1.)    Resistance  in  series  with  a  circuit. 

43.  In    a    constant-potential    system    with    impressed 
E.M.F., 


o  =  e.  +/V,  E.  = 


RESISTANCE,  INDUCTANCE,  CAPACITY.  59 

let  the  receiving  circuit  of  impedance 

Z  =  r  —jx,  z  =  Vr2  +  x'2, 

be  connected  in  series  with  a  resistance,  r0  . 
The  total  impedance  of  the  circuit  is  then 

Z  +  r0  =  r  +  r0—jx\ 
hence  the  current  is 


____ 
•"        Z  +  r0       r+r0  -jx          (r  +  r0)2  -f  *2  ' 

and  the  E.M.F.  of  the  receiving  circuit,  becomes 
E  =  IZ  =  ^°  (r  ~J^  =  ^° 


or,  in  absolute  values  we  have  the  following :  — 
Impressed  E.M.F., 

current, 

zr  zr 


V(r  +  ;-0)2  +  x2       -Vz2  + 
E.M.F.  at  terminals  of  receiver  circuit, 


E  =  EnJ       >*  +  *2        . Eo 


Vs2  +  2rr0  +  r02 
difference  of  phase  in  receiver  circuit,  tan  w  =  - ; 

difference  of  phase  in  supply  circuit,  tan  o>0  = 

since  in  general, 

tan  (phase)  =  ^aginary  component  ^ 
real  component 

a.}    If  x  is  negligible  with  respect  to  r,  as  in  a  non-induc- 
tive receiving  circuit, 


1=  -=3_ 

r+  r. 


and  the  current  and  E.M.F.  at  receiver  terminals  decrease 
steadily  with  increasing  r0 . 


60  ALTERNATING-CURRENT  PHENOMENA. 

b.}    If  r  is  negligible  compared  with  x,  as  in  a  wattless 
receiver  circuit, 

7=        E°        ,  £  =  £.          X      - 


or,  for  small  values  of  r0  , 

/=—  °,  ^  =  ^0; 

that  is,  the  current  and  E.M.F.  at  receiver  terminals  remain 
approximately  constant  for  small  values  of  r0,  and  then  de- 
crease with  increasing  rapidity. 

44.  In  the  general  equations,  x  appears  in  the  expres- 
sions for  /  and  E  only  as  xz,  so  that  /  and  E  assume  the 
same  value  when  x  is  negative,  as  when  x  is  positive  ;  or,  in 
other  words,  series  resistance  acts  upon  a  circuit  with  leading 
current,  or  in  a  condenser  circuit,  in  the  same  way  as  upon  a 
circuit  with  lagging  current,  or  an  inductive  circuit. 

For  a  given  impedance,  z,  of  the  receiver  circuit,  the  cur- 
rent /,  and  E.M.F:,  E,  are  smaller,  as  r  is  larger;  that  is, 
the  less  the  difference  of  phase  in  the  receiver  circuit. 

As  an  instance,  in  Fig.  37  is  shown  the  E.M.F.,  E,  at 
the  receiver  circuit,  for  E0  =  const.  =  100  volts,  s  =  1  ohm  ; 
hence  /  =  E,  and  — 

a.)    r0  =  .2  ohm         (Curve    I.) 
b.)    r0  =  .8  ohm         (Curve  II.) 

with  values  of  reactance,  x  =  V^2  —  r2,  for  abscissae,  from 
x  =  +  1.0  to  x  =  —  1.0  ohm. 

As  shown,  /  and  E  are  smallest  for  x  =  0,  r  =  1.0, 
or  for  the  non-inductive  receiver  circuit,  and  largest  for 
x  =  ±  1.0,  r  =  0,  or  for  the  wattless  circuit,  in  which  latter 
a  series  resistance  causes  but  a  very  small  drop  of  potential. 

Hence  the  control  of  a  circuit  by  series  resistance  de- 
pends upon  the  difference  of  phase  in  the  circuit. 

For  r0  =  .8,  and  x  =  0,  x  =  +  .8,  x  =  —  .8,  the  polar 
diagrams  are  shown  in  Figs.  38  to  40. 


RESISTANCE,  INDUCTANCE,  CAPACITY. 


61 


2.)    Reactance  in  series  witJi  a  circuit. 
45.    In  a  constant  potential  system  of  impressed  E.M.F., 


let  a  reactance,  x0  ,  be  connected  in  series  in  a  receiver  cir- 
cuit of  impedance 

Z  =  r  —  jx,  z  =  -\/r2  -|-  x'2. 


IMPRESSED  E.M.F.  CONSTANT,  E0=IOO 
IMPEDANCE  OF  RECEIVER  CIRCUIT  CONSTANT,  Z  -  1.0 


LINE  RESISTANCE  CONSTANT    n  =.2 


3  -  -.4  T-5  '  '.6  T.7  r-8 


Fig.   37.      Variation  of    Voltage  at   Constant   Series  Resistance  with   Phase  Relation   of 
Receiver  Circuit. 

Then,  the  total  impedance  of  the  circuit  is 
Z  -jx0  =  r—j(x  +#e). 


Er   Er0 


Fig.  38. 

and  the  current  is, 
/= 


E 
Fig.  39. 


Z-jx0      r—j(x  +  x0}' 
/hile  the  difference  of  potential  at  the  receiver  terminals 


r—jx 


62  ALTERNATING-CURRENT  PHENOMENA. 

Or,  in  absolute  quantities  :  — 
Current, 

/_ Eo EQ 

•*    ~ 


Vr*  -f-  (x  +  x0)'2      V 'z'1  +  2xx0  -\-  xa2 
E.M.F.  at  receiver  terminals, 


r     /      r'  +  *«       = J^ 

°  V  ra  +  (*  +  *„)*      V**  +  2*.r0  +  *.a  5 


difference  of  phase  in  receiver  circuit, 

x 

tan  <D   =  -  ; 
r 

difference  of  phase  in  supply  circuit, 


a.}  If  JT  is  small  compared  with  r,  that  is,  if  the  receiver 
circuit  is  non-inductive,  /  and  E  change  very  little  for  small 
values  of  x0  ;  but  if  x  is  large,  that  is,  if  the  receiver  circuit 
is  of  large  reactance,  /  and  E  change  much  with  a  change 
of  x0. 

b.}  If  x  is  negative,  that  is,  if  the  receiver  circuit  con- 
tains condensers,  synchronous  motors,  or  other  apparatus 
which  produce  leading  currents  —  above  a  certain  value  of 
x  the  denominator  in  the  expression  of  E,  becomes  <  z,  or 
E  >  E0  ;  that  is,  the  reactance,  x0  ,  raises  the  potential. 

c.)  E  =  E0  ,  or  the  insertion  of  a  series  inductance,  x0  , 
does,  not  affect  the  potential  difference  at  the  receiver  ter- 

minals, if 

^z*-\-2xx0  +  x02  =  2; 
or,  x0  =  —  2  x. 

That  is,  if  the  reactance  which  is  connected  in  series  in 
the  circuit  is  of  opposite  sign,  but  twice  as  large  as  the 
reactance  of  the  receiver  circuit,  the  voltage  is  not  affected, 
but  E  =  E0,I=  E0/z.  If  x0  <  —  2  x,  it  raises,  if  x0  >  —  Zv, 
it  lowers,  the  voltage. 

We  see,  then,  that  a  reactance  inserted  in  series  in 
an  alternating-current  circuit  will  lower  the  voltage  at  the 


RESISTANCE,  INDUCTANCE,  CAPACITY. 


63 


receiver  terminals  only  when  of  the  same  sign  as  the  reac- 
tance of  the  receiver  circuit ;  when  of  opposite  sign,  it  will 
lower  the  voltage  if  larger,  raise  the  voltage  if  less,  than 
twice  the  numerical  value  of  the  reactance  of  the  receiver 
circuit. 

d.}    If  x  =  0,   that    is,   if    the   receiver    circuit    is    non- 
inductive,  the  E.M.F.  at  receiver  terminals  is  : 


=  (!-}-  *)•'*  expanded  by  the  binomial  theorem 


=  nx 


Therefore,  if  x0  is  small  compared  with  r :  — 


That  is,  the  percentage  drop  of  potential  by  the  insertion 
of  reactance  in  series  in  a  non-inductive  circuit  is,  for  small 


Fig.  40. 


values  of  reactance,  independent  of  the  sign,  but  propor- 
tional to  the  square  of  the  reactance,  or  the  same  whether 
it  be  inductance  or  condensance  reactance. 


64 


AL  TERNA  TING-CURRENT  PHENOMENA. 


46.  As  an  instance,  in  Fig.  41  the  changes  of  current, 
/,  and  of  E.M.F.  at  receiver  terminals,  E,  at  constant  im- 
pressed E.M.F.,  E0,  are  shown  for  various  conditions  of  a 
receiver  circuit  and  amounts  of  reactance  inserted  in  series. 

Fig.  41  gives  for  various  values  of  reactance,  x0  (if  posi- 
tive, inductance  —  if  negative,  condensance),  the  E.M.Fs., 
E,  at  receiver  terminals,  for  constant  impressed  E.M.F., 

VOLTS   E  OR  AMPERES  I 


100 

IMPRESSED  E.'M.F!  CONSTANT,  E 
IMPEDANCE  OF  RECEIVER  CIRC.UI 

I.  r=l.o   x=o 

II.  r=.6     X=H-,8 

111.  r=.e   i=-.8 

=160 
r  CONS 

^ 

FAN 
^ 

T.Z 

=  l 

n  1" 

0 

if 

0 

r 

"V 

V 

\ 

U 

o 

J 

\ 

\ 

n 

0 

/ 

\ 

^ 

\ 

i? 

0 

/ 

\ 

/ 

\ 

12 

n 

/ 

/ 

\'l 

"/ 

/ 

\ 

/ 

/ 

. 

X 

/n 

\" 

^, 

^ 

., 

'ill 

X 

/ 

S 

n 

\ 

^> 

\ 

£ 

^ 

/ 

|X 

. 

/ 

- 

0 

\ 

^ 

\ 

|?° 

^x 

' 

Lj 

/ 

x 

/ 

. 

D 

S 

\ 

\ 

^ 

a.  60 
O 

Y/ 

. 

X 

II 

X" 

| 

0 

\ 

so 

10 
Xo  •*•» 

^ 

^ 

^ 

^ 

-^ 

. 

n 

*< 

_-  —  ' 

_~.  — 

--- 

,  — 

—  - 

| 

o 

. 

0 

1 

0 

0    '<! 

HM 

s  t 

s 

h    'J 

TUCJTANCE 

-REACT 

ANC 

E  - 

-t-CONDENSANCE 

Fig.  41. 

E0  =  100  volts,  and  the  following    conditions   of   receiver 
circuit  •—  z=  1  Qj  r  =  1>0>  x=       0  (Curve  j) 

2=1.0,  r=    .6,^=       .8(CurveII.) 
2=  1.0,  r=    .6,  AT=  —  .8  (Curve  III.) 

As  seen,  curve  I  is  symmetrical,  and  with  increasing  x0 
the  voltage  E  remains  first  almost  constant,  and  then  drops 
off  with  increasing  rapidity. 

In  the  inductive  circuit  series  inductance,  or,  in  a  con- 
denser circuit  series  condensance,  causes  the  voltage  to  drop 
off  very  much  faster  than  in  a  non-inductive  circuit. 


RESISTANCE,  INDUCTANCE,  CAPACITY. 


65 


Series  inductance  in  a  condenser  circuit,  and  series  con- 
densance  in  an  inductive  circuit,  cause  a  rise  of  potential. 
This  rise  is  a  maximum  for  x0  =  i  .8,  or,  x0  =  —  x  (the 
condition  of  resonance),  and  the  E.M.F.  reaches  the  value, 
E  =  167  volts,  or,  E  =  E0z]  r.  This  rise  of  potential  by 
series  reactance  continues  up  to  x0  =  il.6,  or,  x0  =  —  %x, 


Fig.  42. 

where  E  =  100  volts  again ;  and  for  x0  >  1.6  the  voltage 
drops  again. 

At  x0  =  ±  -8,  x  =  =f  .8,  the  total  impedance  of  the  circuit 
is  r  —  j  (x  -f  x0}  =  r  =  .6,  x  +  x0  =  0,  and  tan  S>0  =  0  ; 
that  is,  the  current  and  E.M.F.  in  the  supply  circuit  are 
in  phase  with  each  other,  or  the  circuit  is  in  electrical 
resonance. 


\ 


Fig.  43. 

Since  a  synchronous  motor  in  the  condition  of  efficient 
working  acts  as  a  condensance,  we  get  the  remarkable  result 
that,  in  synchronous  motor  circuits,  choking  coils,  or  reactive 
coils,  can  be  used  for  raising  the  voltage. 

In  Figs.  42  to  44,  the  polar  diagrams  are  shown  for  the 
conditions  — 

E0  =  100,  x0  =  .6,  x  =       0  .  (Fig.  42)  E  =    85.7 

x  =  +  .8  (Fig.  43)  E  =    65.7 

(Fig.  44)  E  =  158.1 


66 


ALTERNA TING-CURRENT  PHENOMENA. 


47.  In  Fig.  45  the  dependence  of  the  potential,  E,  upon 
the  difference  of  phase,  oi,  in  the  receiver  circuit  is  shown 
for  the  constant  impressed  E.M.F.,  E0  =  100  ;  for  the  con- 
stant receiver  impedance,  z  =  1.0  (but  of  various  phase 
differences  to),  and  for  various  series  reactances,  as  follows  : 

x0  =    .2  (Curve  I.) 

x0  =    .6  (Curve  II.) 

x0  =    .8  (Curve  III.) 

xo  =  1.0  (Curve  IV.) 

Xo  =  1.6  (Curve  V.) 

x0  =  3.2  (Curve  VI.) 


Fig.  44. 

Since  z  =  1.0,  the  current,  /,  in  all  these  diagrams  has 
the  same  value  as  E. 

In  Figs.  46  and  47,  the  same  curves  are  plotted  as  in 
Fig.  45,  but  in  Fig.  46  with  the  reactance,  .*•,  of  the  receiver 
circuit  as  abscissas ;  and  in  Fig.  47  with  the  resistance,  r,  of 
the  receiver  circuit  as  abscissae. 

As  shown,  the  receiver  voltage,  E,  is  always  lowest  when 
x0  and  x  are  of  the  same  sign,  and  highest  when  they  are 
of  opposite  sign. 

The  rise  of  voltage  due  to  the  balance  of  x0  and  x  is  a 
maximum  for  x0=  +1.0,  x  =  —  1.0,  and  r  =  0,  where 


RESISTANCE,   INDUCTANCE,    CAPACITY. 


L    Q. 4— PHASE     D FFERENCE    IN    CONSUMER    SIR    UIT 


l-90  80   70  bO  50  40  30  20  10   0   10  20  30  10  50  60   70  bO  90  OEUHE 

fig.  45.     Variation  of  Voltage  at  Constant  Series  Reactance  with  Phase  Angle  of 
Receiver  Circuit. 


Fig.  46.     Variation  of  Voltage  at  Constant  Series  Reactance  with  Reactance  of 
Receiver  Circuit. 


68 


AL  TERN  A  TING-CURRENT  PHENOMENA. 


E  =  oo  ;  that  is,  absolute  resonance  takes  place.  Obvi- 
ously, this  condition  cannot  be  completely  reached  in 
practice. 

It  is  interesting  to  note,  from  Fig.  47,  that  the  largest 
part  of  the  drop  of  potential  due  to  inductance,  and  rise  to 
condensance  —  or  conversely  —  takes  place  between  r  =  1.0 
and  r  =  .9  ;  or,  in  other  words,  a  circuit  having  a  power 


Volts  E 
or  Amperes  I. 
160 
150 
140 
130 
120 
110 
100 
90 
80 
70 


sfl 


Fig.  47.     Variation  of  Voltage  at  Constant  Series  Reactance  with  Resistance  of 
Receiver  Circuit. 

factor  cos  &  =  .9,  gives  a  drop  several  times  larger  than  a 
non-inductive  circuit,  and  hence  must  be  considered  as 
an  inductive  circuit. 

3.)  Impedance  in  series  witJi  a  circuit. 
48.  By  the  use  of  reactance  for  controlling  electric 
circuits,  a  certain  amount  of  resistance  is  also  introduced, 
due  to  the  ohmic  resistance  of  the  conductor  and  the  hys- 
teretic  loss,  which,  as  will  be  seen  hereafter,  can  be  repre- 
sented as  an  effective  resistance. 


RESISTANCE,  INDUCTANCE,  CAPACITY.  69 

Hence  the  impedance  of  a  reactive  coil  (choking  coil) 
may  be  written  thus  :  — 

&Q  =  ro        JXoi  ZQ  =  V  f0    -j-  Xo  , 

where  r0  is  in  general  small  compared  with  x0. 
From  this,  if  the  impressed  E.M.F.  is 


E0  =  e0  +je0'>  E0  =  Ve02  +  e0'2 

and  the  impedance  of  the  consumer  circuit  is 

we  get  the  current,     /= ^-  = -. — 

and  the  E.M.F.  at  receiver  terminals, 

.  .  °   7     \      7  "°  (r     \     *-\  //„_!_   „  \  ' 

•^I^o  \r  ~T  '  o)          J  \*-  ~T  •*<>/ 

Or,  in  absolute  quantities, 
the  current  is, 

~\/(r  -f-  roy2  -|-  (x  -j-  ^;0)2       V^2  +  z02  +  2  (rr0 

the  E.M.F.  at  receiver  terminals  is, 

E0z  E0z 


V(r  +  r0)'2  +  (x  +  xoy         V^2  +  Z0*  +  2 
the  difference  of  phase  in  receiver  circuit  is, 

x 

tan  oi  =  -  ; 
r 

and  the  difference  of  phase  in  the  supply  circuit  is, 


49.  In  this  case,  the  maximum  drop  of  potential  will  not 
take  place  for  either  x  =  0,  as  for  resistance  in  series,  or 
for  r  =  0,  as  for  reactance  in  series,  but  at  an  intermediate 
point.  The  drop  of  voltage  is  a  maximum  ;  that  is,  E  is  a 
minimum  if  the  denominator  of  E  is  a  maximum  ;  or,  since. 
zy  z0,  r0,  x0  are  constant,  if  rr0  +  xx0  is  a  maximum,  that  is, 
since  x  =  ~Vz2  —  r2,  if  rr0  -f-  x0  ~\/z2  —  r2  is  a  maximum. 


70 


AL  TERN  A  TING    CURRENT-PHEXOMENA. 


A  function,  f  =  rr0  -+-  x0  V^2  —  r2  is  a  maximum  when 
its  differential  coefficient  equals  zero.  For,  plotting  f  as 
curve  with  r  as  abscissae,  at  the  point  where  f  is  a  maxi- 
mum or  a  minimum,  this  curve  is  for  a  short  distance 
horizontal,  hence  the  tangens-function  of  its  tangent  equals 
zero.  The  tangens-function  of  the  tangent  of  a  curve,  how- 
ever, is  the  ratio  of  the  change  of  ordinates  to  the  change 
of  abscissae,  or  is  the  differential  coefficient  of  the  func- 
tion represented  by  the  curve. 


/ 

/ 

/ 

/ 

^ 

/ 

/ 

^^«- 

,  " 

•*^ 

'"^— 

^^~ 

Z^ 

£L 

,~-— 

—  ' 

_---* 

/ 

/ 

^__ 

•    • 

•~~  ^ 

.  • 

^  —  • 

,--- 

J^- 

~~    - 

SiL 

9- 

<-* 

I. 

.9 

.8 

Tf 

.0 

J 

.4 

.3 

.2 

., 

- 

-.1  - 

-.2 

-.3   - 

-.4  - 

-•}  ' 

-.fi 

-.? 

-.* 

2J 

Off.  48. 

Thus  we  have  :  — 

f  =  rr0  +  *0  Vs2  —  r2  =  maximum  or  minimum,  if 


Differentiating,  we  get  :  — 


RESISTANCE,  INDUCTANCE,  CAPACITY. 


71 


That  is,  the  drop  of  potential  is  a  maximum,  if  the  re- 
actance factor,  x I r,  of  the  receiver  circuit  equals  the  reac- 
tance factor,  *0/r0,  of  the  series  impedance. 


Fig.  49. 


''o 
Fig.  50. 


50.  As  an  example,  Fig.  48  shows  the  E.M.F.,  E, 
at  the  receiver  terminals,  at  a  constant  impressed  E.M.F., 
E0  =  100,  a  constant  impedance  of  the  receiver  circuit, 
s  =  1.0,  and  constant  series  impedances, 

Z0=    .S-/.4  (Curve  I.) 

Z0  =  1.2  —  / 1.6         (Curve  II.) 
as  functions  of  the  reactance,  x,  of  the  receiver  circuit. 


Fig.  51. 

Figs.    49  to  51  give  the  polar  diagram  for  E0  =  100, 
x  =  .95,  x  =  0,  x  =  -  .95,  and  Z0  =  .3  -/  .4. 


72  ALTERNATING-CURRENT  PHENOMENA. 

4.)    Compensation   for   Lagging    Currents    by    Shunted 
Condensance. 

51.  We  have  seen  in  the  latter  paragraphs,  that  in  a 
constant  potential  alternating-current  system,  the  voltage 
at  the  terminals  of  a  receiver  circuit  can  be  varied  by  the 
use  of  a  variable  reactance  in  series  to  the  circuit,  without 
loss  of  energy  except  the  unavoidable  loss  due  to  the 
resistance  and  hysteresis  of  the  reactance;  and  that,  if 
the  series  reactance  is  very  large  compared  with  the  resis- 
tance of  the  receiver  circuit,  the  current  in  the  receiver 
circuit  becomes  more  or  less  independent  of  the  resis- 
tance,—  that  is,  of  the  power  consumed  in  the  receiver 


Fig.  52. 


circuit,  which  in  this  case  approaches  the  conditions  of  a 
constant  alternating-current  circuit,  whose  current  is. 

/=  —      "        .   or  approximately,  /  =  —  °  . 


This  potential  control,  however,  causes  the  current  taken 
from  the  mains  to  lag  greatly  behind  the  E.M.F.,  and 
thereby  requires  a  much  larger  current  than  corresponds 
to  the  power  consumed  in  the  receiver  circuit. 

Since  a  condenser  draws  from  the  mains  a  leading  cur- 
rent, a  condenser  shunted  across  such  a  circuit  with  lagging 
current  will  compensate  for  the  lag,  the  leading  and  the 
lagging  current  combining  to  form  a  resultant  current  more 
or  less  in  phase  with  the  E.M.F.,  and  therefore  propor- 
tional to  the  power  expended. 


RESISTANCE,  INDUCTANCE,  CAPACITY.  73 

In  a  circuit  shown  diagrammatically  in  Fig.  52,  let  the 
non-inductive  receiver  circuit  of  resistance,  r,  be  connected 
in  series  with  the  inductance,  x0 ,  and  the  whole  shunted  by 
a  condenser  of  condensance,  c,  entailing  but  a  negligible  loss 
of  energy. 

Then,  if  E0  =  impressed  E.M.F.,— 

the  current  in  receiver  circuit  is, 


the  current  in  condenser  circuit  is, 

and  the  total  current  is 

— Jxo      Jc 


or,  in  absolute  terms,  I0 


'•=VfeJ+fe-'/; 


while  the  E.M.F.  at  receiver  terminals  is, 
r 


52.  The  main  current,  70,  is  in  phase  with  the  impressed 
E.M.F.,  E0,  or  the  lagging  current  is  completely  balanced, 
or  supplied  by,  the  condensance,  if  the  imaginary  term  in 
the  expression  of  I0  disappears  ;  that  is,  if 


This  gives,  expanded  : 


Hence  the  capacity  required  to  compensate  for  the 
lagging  current  produced  by  the  insertion  of  inductance- 
in  series  to  a  non-inductive  circuit  depends  upon  the  resis- 
tance and  the  inductance  of  the  circuit.  x0  being  constant, 


74  ALTERNATING-CURRENT  PHENOMENA. 

with  increasing  resistance,  r,  the   condensance   has   to   be 
increased,  or  the  capacity  decreased,  to  keep  the  balance. 

r2  4-  r2 
Substituting  c  =     ^/ "  , 

we  get,  as  the  equations  of  the  inductive  circuit  balanced 
by  condensance :  — 


7  = 


r  —  Jxo 

and  for  the  power  expended  in  the  receiver  circuit  :  — 


that  is,  the  main  current  is  proportional  to  the  expenditure 
of  power. 

For  r  =  0  we  have  c  =  x0,  or  the  condition  of  balance. 

Complete  balance  of  the  lagging  component  of  current 
by  shunted  capacity  thus  requires  that  the  condensance,  <:, 
be  varied  with  the  resistance,  r;  that  is,  with  the  varying 
load  on  the  receiver  circuit. 

In  Fig.  53  are  shown,  for  a  constant  impressed  E.M.F., 
E0  =  1000  volts,  and  a  constant  series  reactance,  x0  =  100 
ohms,  values  for  the  balanced  circuit  of, 

current  in  receiver  circuit  (Curve  I.), 
current  in  condenser  circuit  (Curve  II.), 
current  in  main  circuit  (Curve  III.), 

E.M.F.  at  receiver  terminals  (Curve  IV.), 

with  the  resistance,  r,  of  the  receiver  circuit  as  abscissae. 


RESISTANCE,   INDUCTANCE,    CAPACITY. 


75 


IMPRESSED  E.M.F.  CONSTANT,  E0  =  IOOO  VOLTS. 
SERIES    REACTANCE   CONSTANT,  X0=  IOO  OHMS. 
VARIABLE  RESISTANCE  IN  RECEIVER  CIRCUIT. 
BALANCED  BY  VARYING  THE  SHUNTED  CONDENSANCE, 

I.    CURRENT  IN  RECEIVER  CIRCUIT. 

II.  CURRENT  IN  CONDENSER  CIRCUIT. 

III.  CURRENT  IN  MAIN  CIRCUIT. 
JV.   E.M.F.  AT  RECEIVER  CIRCUIT. 


100  / 


r.  OF  RECEIVER 


CIRCUIT    OHMS 


10  20  30  40  50  60  70  80  90  100  110  120  130  HO  150  160  170  180  190  200 

Fig.  53.    Compensation  of  Lagging  Currents  in  Receiving  Circuit  by  Variable  Shunted 
Condensance. 


53.  If,  however,  the  condensance  is  left  unchanged, 
c  =  x0  at  the  no-load  value,  the  circuit  is  balanced  for  r  =  0, 
but  will  be  overbalanced  for  r  >  0,  and  the  main  current 
will  become  leading. 

We  get  in  this  case  :  — 


r-jx 


The  difference  of  phase  in  the  main  circuit  is,  — 


tan  u>0  = , 

«0 


which  is  =  0. 


76 


ALTERNA TING-CURRENT  PHENOMENA. 


when  r  =  0  or  at  no  load,  and  increases  with  increasing 
resistance,  as  the  lead  of  the  current.  At  the  same  time, 
the  current  in  the  receiver  circuit,  7,  is  approximately  con- 
stant for  small  values  of  r,  and  then  gradually  decreases. 


IMPRESSED  E.M.F.  CONSTANT,  EO—IOOO  VOLTS. 

SERIES   REACTANCE    CONSTANT,  Xt,  -<OO  OHMS. 
SHUNTED  CONDENSANCE  CONSTANT,    C=  IOO  OH 
VARIABLE  RESISTANCE.  IN  RECEIVER  CIRCUIT- 
•(.CURRENT  IN  RECEIVER  CIRCUIT. 
II.  CURRENT  IN  CONDENSER  C  RCUIT. 
III.  CURRENT  IN  MA  N  CIRCUIT. 
IV.E.M.F.  AT  RECEIVER  CIRCUIT. 

MS. 

voi 

- 

ii. 

?00 

"—  •-. 

~~~, 

^^. 

^ 

\. 

.  

„—  

^^ 

-r_-~ 

-x 

^ 

_^- 

L-* 

rnn 

^ 

' 

^ 

•^ 



soo 

IV, 

/ 

** 

""--^ 

•^-^ 

% 

-—  -*. 

-^—  ~, 

300 

/ 

/ 

/ 

RESISTANCE  r—  OF  RECEIVER  CIRCUIT,  OHMS. 

2 

MINI 

JO     20    80    40     50    60    70    80    90    100  110  120  '  130  140  150  100  170  1 

JO   190  200  OHMS 

Fig.  54. 

In  Fig.  54  are  shown  the  values  of  /,  71}  70,  7f,  in  Curves 
I.,  II.,  III.,  IV.,  similarly  as  in  Fig.  50,  for  E0  =  1000  volts, 
c  =  x  =  100  ohms,  and  r  as  abscissas. 


5.)    Constant  Potential —  Constant  Current   Transformation. 

54.  In  a  constant  potential  circuit  containing  a  large 
and  constant  reactance,  x0,  and  a  varying  resistance,  r,  the 
current  is  approximately  constant,  and  only  gradually  drops 
off  with  increasing  resistance,  r,  —  that  is,  with  increasing 
load,  —  but  the  current  lags  greatly  behind  the  E.M.F.  This 
lagging  current  in  the  receiver  circuit  can  be  supplied  by  a 
shunted  condensance.  Leaving,  however,  the  condensance 
constant,  c  =  x0,  so  as  to  balance  the  lagging  current  at  no 


RESISTANCE,  INDUCTANCE,  CAPACITY.  . 


77 


load,  that  is,  at  r  =  0,  it  will  overbalance  with  increasing 
load,  that  is,  with  increasing  r,  and  thus  the  main  current 
will  become  leading,  while  the  receiver  current  decreases 
if  the  impressed  E.M.F.,  E0,  is  kept  constant.  Hence,  to 
keep  the  current  in  the  receiver  circuit  entirely  constant,  the 
impressed  E.M.F.,  E0,  has  to  be  increased  with  increasing 
resistance,  r;  that  is,  with  increasing  lead  of  the  main  cur- 
rent. Since,  as  explained  before,  in  a  circuit  with  leading 
current,  a  series  inductance  raises  the  potential,  to  maintain 
the  current  in  the  receiver  circuit  constant  under  all  loads,, 
an  inductance,  x^ ,  inserted  in  the  main  circuit,  as  shown  ia 
the  diagram,  Fig.  55,  can  be  used  for  raising  the  potential 
E0,  with  increasing  load. 


Fig.  55. 


Let  — 


be  the  impressed  E.M.F.  of  the  generator,  or  of  the  mains, 
and  let  the  condensance  be  xc  =  x0\  then  —  • 
Current  in  receiver  circuit, 


r  —jx0 


current  in  condenser  circuit, 

T 

/I  =  — 


X0 


Hence,  the  total  current  in  main  line  is 


r—  x        x 


78  A  L  TERN  A  TING-CURRENT  PHENOMENA. 

and  the  E.M.F.  at  receiver  terminals, 

r  —JXo 
E.M.F.  at  condenser  terminals, 

E.M.F.  consumed  in  main  line, 
hence,  the  E.M.F.  at  generator  is 


and  conversely  the  E.M.F.  at  condenser  terminals, 


current  in  receiver  circuit, 
7 


r  —jx0       r  (x0  —  xj  —jx?  ' 

This  value  of  /  contains  the  resistance,  r,  only  as  a  fac- 
tor to  the  difference,  x0  —  x^\  hence,  if  the  reactance,  ;r2  , 
is  chosen  =  x0  ,  r  cancels  altogether,  and  we  find  that  if 
#2  =  *0,  the  current  in  the  receiver  circuit  is  constant, 

/-/A, 

X0 

and  is  independent  of  the  resistance,  r  ;  that  is,  of  the  load. 

Thus,  by  substituting  xz  =  x0,  we  have, 
Impressed  E.M.F.  at  generator, 

E<i  =  <?2  +  Je*'i  Ez  =  V^22  +  ^2'  2  =  constant  ; 

current  in  receiver  circuit, 

/    =j%L,  7  =  ^?  =  constant; 

x0  xa 

E.M.F.  at  receiver  circuit, 

E  =  Ir=jE-^-,        E  ~  ^^,  or  proportional  to  load  r; 

' 


RESISTANCE,  INDUCTANCE,  CAPACITY.  79 

E.M.F.  at  condenser  terminals, 


E*     1  +/  -    ,  £0=  ^2  V  1  +    -    ,  hence  >  E,  • 


.V 

current  in  condenser  circuit, 


main  current, 


r 


°       *.(*.+./>)  ' 

(  proportional  to  the  load, 
T       JZI<L  f      1  ,    .         ,  .  , 

/o  =  —  V  '       J  r»     anC^     ln     Pnase     Wlt" 

°  X°        (  E.M.F.,  Ez  . 

The  power  of  the  receiver  circuit  is, 


the  power  of  the  main  circuit, 

f0Ez  =     2  r  ,  hence  the  same. 
*02 

55.    This  arrangement  is  entirely  reversible  ;  that  is, 
if  Ez  =  constant,  /    =  constant  ;  and 
if  I0    =  constant,  E  =  constant. 

In  the  latter  case  we  have,  by  expressing  all  the  quanti- 
ties by  70  :  — 
Current  in  main  line, 

I0   =  constant; 
E.M.F.  at  receiver  circuit, 

E  =  I0x9  =  constant  ; 
current  in  receiver  circuit, 

/    =f0  —  ,  proportional  to  the  load  -; 
current  in  condenser  circuit, 


80  AL  TERNA  TING-CURRENT  PHENOMENA. 

E.M.F.  at  condenser  terminals, 


Impressed  E.M.F.  at  generator  terminals, 

x  2  1 

£2  =  —I0  ,  or  proportional  to  the  load  -  . 

From  the  above  we  have  the  following  deduction  : 

Connecting  two  reactances  of  equal  value,  x0,  in  series 
to  a  non-inductive  receiver  circuit  of  variable  resistance,  r, 
and  shunting  across  the  circuit  from  midway  between  the 
inductances  by  a  capacity  of  condensance,  xc  =  x0,  trans- 
forms a  constant  potential  main  circuit  into  a  constant  cur- 
rent receiver  circuit,  and,  inversely,  transforms  a  constant 
current  main  circuit  into  a  constant  potential  receiver  cir- 
cuit. This  combination  of  inductance  and  capacity  acts  as 
a  transformer,  and  converts  from  constant  potential  to  con- 
stant current  and  inversely,  without  introducing  a  displace- 
ment of  phase  between  current  and  E.M.F. 

It  is  interesting  to  note  here  that  a  short  circuit  in  the 
receiver  circuit  acts  like  a  break  in  the  supply  circuit,  and  a 
break  in  the  receiver  circuit  acts  like  a  short  circuit  in  the 
supply  circuit. 

As  an  instance,  in  Fig.  56  are  plotted  the  numerical 
values  of  a  transformation  from  constant  potential  of  1,000 
volts  to  constant  current  of  10  amperes. 

Since  E^  =  1,000,  7=10,  we  have  :  x0  =  100  ;  hence 
the  constants  of  the  circuit  are  :  — 

E*  =  1000  volts  ; 

7    =  10  amperes  ; 

E  —  10  r,  plotted  as  Curve  I.,  with  the  resistances,  r,  as  abscissa;; 

E0  =  1000  1/1  +  I  —  Y  plotted  as  Curve  II.  ; 
»'         V  100  y 

7t  =  10  i/1  +  (  -£-Y,  plotted  as  Curve  III.- 
V  ^-^^  J 

70  =  .1  r,  plotted  as  Curve  IV. 


RESISTANCE,  INDUCTANCE,  CAPACITY. 


81 


56.  In  practice,  the  power  consumed  in  the  main  circuit 
will  be  larger  than  the  power  delivered  to  the  receiver  cir- 
cuit, due  to  the  unavoidable  losses  of  power  in  the  induc- 
tances and  condensances. 


u 

13 
12 

11 
10 
9 

j« 

|.7 

6 
6 
1 
3 
2 
1 

— 

CURRENT  IN  RECEIVER  CIRCUIT  CONSTANT, 
IMPR£SSED  E.M,  F.CONSTANT,  E8=IOOO  VOL 
2    REACTANCES   OFOTo  =IOO  OHMS  EACH,  SH 
THE  CONDENSANCE,  ZC  =  IOO  OHMS. 
VARIABLE  RES  STANCE  IN  RECEIVER  CIRCUI 
1     E.M.F.  AT  RECEIVER  C  RCUIT. 
1         II     E.M-F.  AT  CONDENSER  CIRCUIT. 
Ill     CURRENT  IN  CONDENSER  CIRCUIT. 
IV     CURRENT  IN  MAIN  LINE 
V     CURRENT  IN  MAIN  LINE  INCLUDING  tC 
VI     EFFICIENCY  OF  TRANSFORMATION, 

1^10  AMPERES       1 

rs.                       '~ 

UNTED  IN  THEIR  MID: 

LE 

BY 

r. 

ou 

1100 

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1300 

SSES 

,-- 

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1200 

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«-—  — 

VI 

"^ 

X 

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~ 

^ 

-^ 

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son 

X 

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581 

^ 

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^xi 

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700 

/ 

x^ 

X-1 

/ 

^ 

600/ 

^ 

' 

^  \ 

•^ 

4 

^ 

-'' 

^ 

•^ 

L 

--'• 

^ 

^ 

1 

}joo 

,x 

X 

^ 

IMO 

^x 

^> 

^ 

100 

,. 

^ 

^ 

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ST* 

NCE 

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r  c 

F   R 

ECE 

VE 

H  Cl 

RCL 

IT, 

OH 

AS 

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.0    1 

(1  1 

I.  V- 

1   V 

1)  2 

»  () 

HM8 

F/3.  50.    Constant-Potential  —  Constant-Current  Transformation. 

Let  — 

ri  =  2  ohms  =  effective  resistance  of  condensance  ; 

r0  =  3  ohms  =  effective  resistance  of  each  of  the  inductances. 

We  then  have  :  — 

Power  consumed  in  condensance,  I*  r±  =  200  +  .02  r2 ; 
power  consumed  by  first  inductance,  72  r0  =  300  ; 
power  consumed  by  second  inductance,  /02r0  =  .03  r*. 
Hence,  the  total  loss  of  energy  is  500  +  -05  r2 ; 
output  of  system,  /2  r  =  100  r 

input,  500  +  100  r  -\ 

effidenCy'  500  +  1W  M 

It  follows  that  the  main  current,  f0,  increases  slightly 
by  the  amount  necessary  to  supply  the  losses  of  energy 
in  the  apparatus. 


82  ALTERNATING-CURRENT  PHENOMENA. 

This  curve  of  current,  I0,  including  losses  in  transforma- 
tion, is  shown  in  dotted  lines  as  Curve  V.  in  Fig.  56  ;  and 
the  efficiency  is  shown  in  broken  line,  as  Curve  VI.  As 
shown,  the  efficiency  is  practically  constant  within  a  wide 
range. 


RESISTANCE   OF  TRANSMISSION  LINES. 


CHAPTER   IX. 

RESISTANCE   AND    REACTANCE    OF    TRANSMISSION   LINES. 

57.  In  alternating-current  circuits,  E.M.F.  is  consumed 
in  the  feeders  of  distributing  networks,  and  in  the  lines  of 
long-distance  transmissions,  not  only  by  the  resistance,  but 
also  by  the  reactance,  of  the  line.  The  E.M.F.  consumed  by 
the  resistance  is  in  phase,  while  the  E.M.F.  consumed  by  the 
reactance  is  in  quadrature,  with  the  current.  Hence  their 
influence  upon  the  E.M.F.  at  the  receiver  circuit  depends 
upon  the  difference  of  phase  between  the  current  and  the 
E.M.F.  in  that  circuit.  As  discussed  before,  the  drop  of 
potential  due  to  the  resistance  is  a  maximum  when  the 
receiver  current  is  in  phase,  a  minimum  when  it  is  in 
quadrature,  with  the  E.M.F.  The  change  of  potential  due 
to  line  reactance  is  small  if  the  current  is  in  phase  with 
the  E.M.F.,  while  a  drop  of  potential  is  produced  with  a 
lagging,  and  a  rise  of  potential  with  a  leading,  current  in 
the  receiver  circuit. 

Thus  the  change  of  potential  due  to  a  line  of  given  re- 
sistance and  inductance  depends  upon  the  phase  difference 
in  the  receiver  circuit,  and  can  be  varied  and  controlled 
by  varying  this  phase  difference ;  that  is,  by  varying  the 
admittance,  Y  =  g  -f  jb,  of  the  receiver  circuit. 

The  conductance,  gy  of  the  receiver  circuit  depends  upon 
the  consumption  of  power,  —  that  is,  upon  the  load  on  the 
circuit,  —  and  thus  cannot  be  varied  for  the  purpose  of  reg- 
ulation. Its  susceptance,  b,  however,  can  be  changed  by 
shunting  the  circuit  with  a  reactance,  and  will  be  increased 
by  a  shunted  inductance,  and  decreased  by  a  shunted  con- 
densance.  Hence,  for  the  purpose  of  investigation,  the 


84  ALTERNATING-CURRENT  PHENOMENA. 

receiver  circuit  can  be  assumed  to  consist  of  two  branches, 
a  conductance,  g,  —  the  non-inductive  part  of  the  circuit,  — 
shunted  by  a  susceptance,  b,  which  can  be  varied  without 
expenditure  of  energy.  The  two  components  of  current 
can  thus  be  considered  separately,  the  energy  component  as 
determined  by  the  load  on  the  circuit,  and  the  wattless 
component,  which  can  be  varied  for  the  purpose  of  regu- 
lation. 

Obviously,  in  the  same  way,  the  E.M.F.  at  the  receiver 
circuit  may  be  considered  as  consisting  of  two  components, 
the  energy  component,  in  phase  with  the  current,  and 
the  wattless  component,  in  quadrature  with  the  current. 
This  will  correspond  to  the  case  of  a  reactance  connected 
in  series  to  the  non-inductive  part  of  the  circuit.  Since  the 
effect  of  either  resolution  into  components  is  the  same  so 
far  as  the  line  is  concerned,  we  need  not  make  any  assump- 
tion as  to  whether  the  wattless  part  of  the  receiver  circuit 
is  in  shunt,  or  in  series,  to  the  energy  part. 

Let— 

Z0  =  r0  —,jx0  =  impedance  of  the  line  ; 

z0   =  Vr02  +  ^2; 
Y  =  g  -\-jb    =  admittance  of  receiver  circuit; 

y  =  VFTT2; 

E0  =  e0  -f  /<?</  =  impressed  E.M.F.  at  generator  end  of  line  ; 

E0  = 
E  =  e    +/<?'   =  E.lVf.F.  at  receiver  end  of  line  ; 


E  = 


I0   =  i0  -\-jio    =  current  in  the  line  ; 

I0  =  Vtf  +  4". 
The  simplest  condition  is  the  non-inductive  circuit. 

1.)    Non-inductive  Receiver  Circuit  Sripplied  over  an 

Inductive  Line. 

58.    In  this  case,  the  admittance  of  the  receiver  circuit 
is  Y  =  g,  since  b  =  0. 


RESISTANCE   OF  TRANSMISSION  LINES.  85 

We  have  then  — 

current,  70  =  Eg; 

impressed  E.M.F.,  E0  =  E  +  Z0  70  =  E  (1  +  Z.g). 

Hence  — 
E.M.F.  at  receiver  circuit, 

=  \^Z0g~  \-\-gr.-jgxJ 
current,  70  =  JA|_  =  ^          . 

Hence,  in  absolute  values  — 
E.M.F.  at  receiver  circuit,  E 

current,  70  : 


The  ratio  of  E.M.Fs.  at  receiver  circuit  and  at  genera- 
tor, or  supply  circuit,  is  — 


and  the  power  delivered  in  the  non-inductive  receiver  cir- 
cuit, or 

output,  P  =  I0  E  = 


As  a  function  of  g,  and  with  a  given  Eot  r0,  and  x0,  this 
power  is  a  maximum,  if  — 


that  is  — 

-l+^-V^+^^^O; 
hence  — 

conductance  of  receiver  circuit  for  maximum  output, 


Vr02  +  V       ^o 
Resistance  of  receiver  circuit,     rm  =  —  =  z0  ; 


86  AL  TERNA  TING-CURRENT  PHENOMENA. 

and,  substituting  this  in  P  — 

Maximum  output,         Pm  = 2 =  —       g — 

and  — 

ratio  of  E.M.F.  at  receiver  and  at  generator  end  of  line, 

am  =  -=r  = 


efficiency, 


That  is,  the  output  which  can  be  transmitted  over  an 
inductive  line  of  resistance,  r0  ,  and  reactance,  x0  ,  —  that  is, 
of  impedance,  z0  ,  —  into  a  non-inductive  receiver  circuit,  is 
a  maximum,  if  the  resistance  of  the  receiver  circuit  equals 
the  impedance  of  the  line,  r  =  z0)  and  is  — 


The  output  is  transmitted  at  the  efficiency  of 


and  with  a  ratio  of  E.M.Fs.  of 

1 


59.  We  see  from  this,  that  the  maximum  output  which 
can  be  delivered  over  an  inductive  line  is  less  than  the 
output  delivered  over  a  non-inductive  line  of  the  same 
resistance  —  that  is,  which  can  be  delivered  by  continuous 
currents  with  the  same  generator  potential. 

In  Fig.  57  are  shown,  for  the  constants 

E0  =  1000  volts, 

Zg  =  2.5  —  6/ ;  that  is,  r,  =  2.5  ohms,  x0  —  6  ohms,  z0  =  6.5  ohms, 

with  the  current  I0  as  abscissae,  the  values  — 


RESISTANCE    OF   TRANSMISSION  LINES. 


87 


E.M.F.  at  Receiver  Circuit,  E,  (Curve  I.) ; 

Output  of  Transmission,  P,  (Curve  II.) ; 

Efficiency  of  Transmission,  (Curve  III.). 

The  same  quantities,  E  and  P,  for  a  non-inductive  line  of 
resistance,  r0  =  2.5  ohms,  x0  =  0,  are  shown  in  Curves  IV., 
V.,  and  VI. 


SUPFUED'OVER  INDUCTIVE  LINE  OF  IMPEDAN 
AND  OVER  NON-INDUCTIVE  LII^E  OF  RESISTAr. 

T0  =  2.5 
CURVE  1.    E.  M.  F.  AT  RECEIVER  CIRCUIT,  INDUCTIVE  LI 

3E 

CE 
SE 

UK 

100 
90 
80 
70 

CO 
50 
40 
30 
40 
10 

^^x- 

---1 

.  

ii      V.           11          ii             ii             ii   NON-INDUCTIVE      » 

^ 

x'. 

0 

t 

VI 

" 

" 

sos 

NDL 

CTIV 

/ 

•4 

"?" 

z 

5 
-S- 
o 

/ 

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fc 

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5 

0 

/ 

o 

^ 

/ 

| 

co 

/ 

jjj 

0 

/, 

/* 

*~^ 

IIMl' 

m 

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"^ 

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// 

/ 

<> 

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m 

^^ 

^ 

^ 

^^^ 

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\ 

gpj 

JQJ 

/ 

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^^ 

<^ 

^~, 

f^ 

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B3 

TOO 

/ 

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>> 

/r 

5 

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jj^ 

300 

^ 

Xs- 

x 

S 

x 

\ 

.-»i  )  1 

"~  —  . 

no 



/ 

\ 

\ 

\ 

40j 

wo 

/ 

s 

x\ 

ai-r 

.300 

/ 

s 

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L'OO 

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100 

1 

cu 

^RE 

NT 

N    L 

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AMF 

ERE 

s 

\ 

10     20     30     40     50     60     70    80 
Fig.  57.    Non-inductive  Receiver  Circuit  Supplied  Over  Inductive  Line. 

2.)    Maximum  Power  Supplied  over  an  Inductive  Line. 

60.  If  the  receiver  circuit  contains  the  susceptance,  b, 
in  addition  to  the  conductance,  g,  its  admittance  can  be 
written  thus :  — 

Then  — 
current, 
Impressed  E.M.F., 


/„  =  E  Y; 
E0  =  E  +  I0Z0  ==  E  (1  +  KZ0). 


88  AL  TERNA  TING-CURRENT  PHENOMENA. 

Hence  — 
E.M.F.  at  receiver  terminals, 


1  +  FZ0        (1  +  r.g  +  x.S)  -  J  (x.g  -  r.6)' 
current, 


or,  in  absolute  values  — 
E.M.F.  at  receiver  circuit, 


V(l  +  r.f  +  x,bf  +  (x.g  -  r. 
current, 


=  E  J  _  jr2  +  ^2  _  . 

°  V  (i  +  rog  +  Xoby  +  (Xog  -  r0t>y' 


ratio  of  E.M.Fs.  at  receiver  circuit  and  at  generator  circuit, 
E  1 


and  the  output  in  the  receiver  circuit  is, 
P=E*g=  E?o?g. 

61.  a.)  Dependence  of  the  output  upon  the  susceptance  of 
the  receiver  circuit. 

At  a  given  conductance,  g,  of  the  receiver  circuit,  its 
output,  P  =  E?a?g,  is  a  maximum,  if  a2  is  a  maximum  ;  that 
is,  when  — 

/=!=(!  +  r.g  +  x.Vf  +  (x.g  -  r0b? 

is  a  minimum. 

The  condition  necessary  is  — 


or,  expanding,         ,.,  ,N  ,  ,N       A 

5'.    *.  (1  +  rog  +  jf0^)  -  r0  (Xog  -  r0b}  =  0. 

Hence  — 
Susceptance  of  receiver  circuit, 

t=~^^)=~^=  ~b°' 
or  b  +  b0  =  0, 


RESISTANCE   OF  TRANSMISSION  LINES.  89 

that  is,  if  the  sum  of  the  susceptances  of  line  and  of  receiver 
circuit  equals  zero. 

Substituting  this  value,  we  get  — 

ratio  of  E.M.Fs.  at  maximum  output, 


E0       z0  (g 
maximum  output, 

Pl  =  - 


current, 

E0Y  E0  (g 


E0(g-jb0} 


og  -  x0b.}  -J(r0b0 


Io  =  E°  V  (1  +  rog  -  Xob0?  +  (r0b0  +  Xog)*> 
and,  expanding, 

r  =  * 

' 


phase  difference  in  receiver  circuit, 

tan  «  =  *  =  -  A  . 
^  A" 

phase  difference  in  generator  circuit, 


62.    b.}    Dependence  of  the  output  upon  the  conductance 

of  the  receiver  circuit. 

At  a  given  susceptance,  ^,  of  the  receiver  circuit,  its 
output,  P  —  Eo<?g,  is  a  maximum,  if  — 


dP  dl\\ 

-r  =  0,  or  —  I  -  I  =  0, 

dg  d^P] 

)*  +  (Xog  - 


90  ALTERNATING-CURRENT  PHENOMENA. 

that  is,  expanding,  — 

C1  +  r0g  -f  x0  b}2  +  (Xog  —  r0by  —  2g(r0  +  r*g  -f  x*g)  =  0  ; 
or,  expanding, — 

Substituting  this  value  in  the  equation  for  a,  page  88, 
we  get  - 
ratio  of  E.M.Fs., 


power 


As  a  function  of  the  susceptance,  b,  this  power  becomes 
a  maximum  for  dP^j  db  =  0,  that  is,  according  to  §  61,  if  — 
*'--*„. 

Substituting  this  value,  we  get  — 

£=  —  bt>  g  =  So*  y  =  y<n  hence:    Y=  g-\-  jb=  g0  —  jb0\ 

x  =  -  x0  ,  r  =  r0  ,  z  =  z0,  Z  =  r  —  Jx  =  r0  +  jx0  ; 

substituting  this  value,  we  get  — 


ratio  of  E.M.Fs.,  m  . 

power,  ^m  =  i-2-  ; 

that  is,  the  same  as  with  a  continuous-current  circuit  ;  or, 
in  other  words,  the  inductance  of  the  line  and  of  the  receiver 
circuit  can  be  perfectly  balanced  in  its  effect  upon  the 
output. 

63.    As  a  summary,  we  thus  have  : 

The  output   delivered  over  an   inductive  line  of    impe- 


RESISTANCE   OF  TRANSMISSION  LINES.  91 

dance,  Z0  =  r0  —jx0 ,  into  a  non-inductive  receiver  circuit,  is 
a  maximum  for  the  resistance,  r  =  z0,  or  conductance,  g  = 
y0 ,  of  the  receiver  circuit,  or  — 


2  (r.  + 
at  the  ratio  of  potentials, 


With  a  receiver  circuit  of  constant  susceptance,  b,  the  out- 
put, as  a  function  of  the  conductance,  g,  is  a  maximum  for 
the  conductance,  — 

and  is 

EO '  y? 

=  2(^+Vo)' 
at  the  ratio  of  potentials, 


With  a  receiver  circuit  of  constant  conductance,  g,  the 
output,  as  a  function  of  the  susceptance,  b,  is  a  maximum 
for  the  susceptance,  b  =  —  b0,  and  is 


P= 


tffe+JJ?' 

at  the  ratio  of  potentials, 

1 

7o  (£•  +  go)  ' 

The  maximum  output  which  can  be  delivered  over  an  in- 
ductive line,  as  a  function  of  the  admittance  or  impedance 
of  the  receiver  circuit,  takes  place  when  Z  =  r0  -\-jx0,  or 
y=jTo~J6o>  that  is,  when  the  resistance  or  conductance 
of  receiver  circuit  and  line  are  equal,  the  reactance  or  sus- 
ceptance of  the  receiver  circuit  and  line,  are  equal  but  of 
opposite  sign,  and  is,  P  =  E?  /  4  r0 ,  or  independent  of  the 
reactances,  but  equal  to  the  output  of  a  continuous-current 


92 


AL  TERN  A  TING-CURRENT  PHENOMENA. 


circuit  of  equal  line  resistance.  The  ratio  of  potentials  is,  in 
this  case,  a  =  zo  j  2  roi  while  in  a  continuous-current  circuit 
it  is  equal  to  £.  The  efficiency  is  equal  to  50  per  cent. 


.03    .01    .05    .08  ,07     .08    .09  .10    .11    .12    .13    .14    J5    J6    33 

Fig.  58.     Variation  of  the  Potential  in  Line  at  Different  Loads. 

64.    As  an  instance,   in    Fig.   58    are   shown,    for   the 
constants  — 

E0  =  1000  volts,  and  Z0  =  2.5  —  6/;  that  is,  for 

r0  =  2.5  ohms,  x0  =  Gohms,  z0  =  6.5  ohms, 

and  with  the  variable  conductances  as  abscissae,  the  values 
of  the  — 

output,  in  Curve  I.,  Curve  III.,  and  Curve  V. ; 

ratio  of  potentials,  in  Curve  II.,  Curve  IV.,  and  Curve  VI.; 

Curves  I.  and  II.  refer  to  a  non-inductive  receiver 
circuit ; 


RESISTANCE   OF   TRANSMISSION  LINES, 


Curves  III.  and  IV.  refer  to  a  receiver  circuit  of 

constant  susceptance b  =  .142 

Curves  V.  and  VI.  refer  to  a  receiver  circuit  of 

constant  susceptance b  =  —  .142  ; 

Curves  VII.  and  VIII.  refer  to  a  non-inductive  re- 
ceiver circuit  and  non-inductive  line. 

In  Fig.  59,  the  output  is  shown  as  Curve  I.,  and  the 
ratio  of  potentials  as  Curve  II.,  for  the  same  line  constants, 
fora  constant  conductance,  ^-  =  .0592  ohms,  and  for  variable 
susceptances,  b,  of  the  receiver  circuit. 


OUTPUT    P  /NO  RATIO  OF  POTENTIAL  a  t 
SENDING  END  OF  LINE  OF  IMPEDANCE.  Z0 

T  RECEIV1  NG^ND 
=5.5  -3j             

AT 

CON 

TAN 

g=  .  0592 

1    OUTPUT 
II   RATIO  OF  POTENTIALS             — 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\\ 

/ 

\\ 

/ 

I 

/ 

/ 

Ns 

f 

\ 

1 

/ 

\ 

\ 

/ 

\\ 

/ 

5 

/ 

\\ 

/ 

'/ 

\ 

\ 

/ 

7 

\° 

/ 

/ 

\ 

\ 

/ 

P 

\ 

* 

\ 

X 

-<, 

~^_ 

^ 

^« 

'  —  -. 

SUSCEfA 

°T' 

iECE 

IVE 

R  C 

KCU 

IT 

-.3       -.2       -.1  0  +.1       +.2       +.3       +.4 

Fig.  59.     Variation  of  Potential  in  Line  at  Various  Loads. 

3.)   Maximum  Efficiency. 

65.  The  output,  for  a  given  conductance,  g,  of  a  receiver 
circuit,  is  a  maximum  if  b  =  —  b0.  This,  however,  is  gen- 
erally not  the  condition  of  maximum  efficiency. 


94  ALTERNATING-CURRENT  PHENOMENA. 

The  loss  of  energy  in  the  line  is  constant  if  the  current 
is  constant  ;  the  output  of  the  generator  for  a  given  cur- 
rent and  given  generator  E.M.F.  is  a^aximum  if  the  cur- 
rent is  in  phase  with  the  E.M.F.  at  the  generator  terminals. 
Hence  the  condition  of  maximum  output  at  given  loss,  or 
of  maximum  efficiency,  is  — 

tan  £>0  =  0. 
The  current  is  — 


The  current  I0,  is  in  phase  with  the  E.M.F.,  E0,  if  its 
quadrature  component  —  that  is,  the  imaginary  term  —  dis- 
appears, or 

x  +  Xo  =  0. 

This,  therefore,  is  the  condition  of  maximum  efficiency, 


Hence,  the  condition  of  maximum  efficiency  is,  that  the 
reactance  of  the  receiver  circuit  shall  be  equal,  but  of  oppo- 
site sign,  to  the  reactance  of  the  line. 

Substituting  x  =  —  x0,  we  have, 
ratio  of  E.M.Fs., 


power, 


RESISTANCE    OF   TRANSMISSION  LINES. 


95 


and  depending  upon  the  resistance  only,  and  not  upon  the 
reactance. 

This  power  is  a  maximum  if  g  =  g0,  as  shown  before; 
hence,  substituting  g  =  g0,  r  =  r0, 

E  2 

maximum  power  at  maximum  efficiency,  Pm  =  —2— , 

at  a  ratio  of  potentials,         am  —  — -2—  , 

"  ro 

or  the  same  result  as  in  §  62. 


.01  .03    •        .03  .01  .05  .06  .07  .08 

Fig.  60.    Load  Characteristic  of  Transmission  Line. 

In  Fig.  60  are  shown,  for  the  constants  — 
E0  =  1,000  volts, 
Z0  =2.5  —  6/;    r0  =  2.5  ohms,  x0  =  6  ohms,  z0  =  6.5  ohms, 


96  ALTERNATING-CURRENT  PHENOMENA. 

and  with  the  variable  conductances,  g,  of  the  receiver  circuit 
as  abscissae,  the  — 

Output  at  maximum  efficiency,     (Curve  I.)  ; 

Volts  at  receiving  end  of  line,      (Curve  II.)  ; 

Efficiency  =  • ,  (Curve  III.). 

r  +  r0 

4.)    Control  of  Receiver  Voltage  by  Shunted  Snsceptance. 

66.  By  varying  the  susceptance  of  the  receiver  circuit, 
the  potential  at  the  receiver  terminals  is  varied  greatly. 
Therefore,  since  the  susceptance  of  the  receiver  circuit  can 
be  varied  at  will,  it  is  possible,  at  a  constant  generator 
E.M.F.,  to  adjust  the  receiver  susceptance  so  as  to  keep 
the  potential  constant  at  the  receiver  end  of  the  line,  or  to 
vary  it  in  any  desired  manner,  and  independently  of  the 
generator  potential,  within  certain  limits. 

The  ratio  of  E.M.Fs.  is  — 


If  at  constant  generator  potential  E0,  the  receiver  potential 
E  shall  be  constant, 

a  —  constant ; 
hence, 

#2' 
or,  expanding, 


which  is  the  value  of  the  susceptance,  b,  as  a  function  of 
the  receiver  conductance,  —  that  is,  of  the  load,  —  which  is 
required  to  yield  constant  potential,  aE0,  at  the  receiver 
circuit. 

For  increasing  g,  that  is,  for  increasing  load,  a  point  is 
reached,  where,  in  the  expression  — 


b  =  - 


RESISTANCE    OF   TRANSMISSION  LINES. 


97 


the  term  under  the  root  becomes  imaginary,  and  it  thus 
becomes  impossible  to  maintain  a  constant  potential,  aE0. 
Therefore,  the  maximum  output  which  can  be  transmitted 
at  potential  aE0,  is  given  by  the  expression  — 


hence  b  =  —  o0  , 

and      g  =  —  g0  -\- 


the  susceptance  of  receiver  circuit, 
the  conductance  of  receiver  circuit; 


°-  —f»      the  output. 


67.    If  a  =  1,  that  is,  if  the  voltage  at  the  receiver  cir- 
cuit  equals  the  generator  potential  — 

P=E*(ty00'-g0). 
If          a  =  1  when  g  =  0,       b  =  0 

when  g  >  0,       b  <  0  ; 
if  a  >  1  when  g  =  0,  or  g  >  0,  b  <  0, 

that  is,  condensance; 
if          a  <  1  when  g  =  0,       b  >  0, 

when  g  =  -  #,  +  \/f  —  ^  -  <V,       ^  =  0  ; 
when^>  -g0  +  V/f  —  ^  -  V,       *  <  0, 


or,  in  other  words,  if  a  <  1,  the  phase  difference  in  the  main 
line  must  change  from  lag  to  lead  with  increasing  load. 

68.    The  value  of  a  giving  the  maximum  possible  output 
in  a  receiver  circuit,  is  determined  by  dP  /  da  =  0 ; 

expanding  :  2  a  (yJL  -  g\  _  f!f'  =  0  ; 

\a  J        a 

hence,  y0  =  2ag0, 

yo  1  Zo 

"  =       =  = 


98  ALTERNATING-CURRENT  PHENOMENA. 

the  maximum  output  is  determined  by  — 

S  ==          So     i  =  So  I 

and  is,  P  =  —2-  . 

4  r 

From :  a  =  ^  =  -^-  , 

the    line    reactance,   x0,    can    be    found,   which    delivers    a 
maximum  output  into  the  receiver  circuit  at  the  ratio  of 
potentials,  a, 
and  z0  =  2  r0  a, 

for  a  ==  1, 


If,  therefore,  the  line  impedance  equals  2#  times  the  line 
resistance,  the  maximum  output,  P  =  E*  j  ±  r0,  is  trans- 
mitted into  the  receiver  circuit  at  the  ratio  of  potentials,  a. 

If  z0  =  2  r0,  or  x0  =  r0  V3,  the  maximum  output,  P  = 
£02/4:r0,  can  be  supplied  to  the  receiver  circuit,  without 
change  of  potential  at  the  receiver  terminals. 

Obviously,  in  an  analogous  manner,  the  law  of  variation 
of  the  susceptance  of  the  receiver  circuit  can  be  found  which 
is  required  to  increase  the  receiver  voltage  proportionally  to 
the  load  ;  or,  still  more  generally,  —  to  cause  any  desired 
variation  of  the  potential  at  the  receiver  circuit  indepen- 
dently of  any  variation  of  the  generator  potential,  as,  for  in- 
stance, to  keep  the  potential  of  a  receiver  circuit  constant, 
even  if  the  generator  potential  fluctuates  widely. 

69.  In  Figs.  61,  62,  and  63,  are  shown,  with  the  output, 
P  =  E* g a2,  as  abscissae,  and  a  constant  impressed  E.M.F., 
E0  =  1,000  volts,  and  a  constant  line  impedance,  Z0  = 
2.5  —  6/,  or,  r0  =  2.5  ohms,  x0  =  6  ohms,  z  =  6.5  ohms, 
the  following  values  : 


RATIO'OF  RECEIVER  VOLTAGE  TO  SENDER  VOLTAGE:  d  =I.O 

LINE  IMPEDANCE:  Z0=  a. 5— 6; 

ENERGY  CURRENT  CONSTANT  GENERATOR 

TOTAL  CURRENT 

CURRENT  IN  NON-INDUCTIVE  RECEIVER  CIRCUIT  WITHOUT  COMPENSATION 


OUTPUT]  IN  RECEIVER  CIPJCUIT,   KILOWJATT 
50  60  70  80 

Fig.  61.     Variation  of  Voltage  Transmission  Lines. 


•  . 

RATIO  OF  RECEIVER  VOLTAGE  TO  SENDER  VOLTAGE: 
LINE     MPEDANCE:Z_  =  2.5.—  6J 
\.  ENERGY  CURRENT               CONSTANT  GENRATOR  POT 
II.   REACTIVE  CURRENT 
III.  TOTAL  CURRENT 
IV.  POTENTIAL  IN  NON-INDUCTIVE  CIRCUIT  WITHOUT  C 

~|Tt-MJJ   MINI 

a  =.7 
:NTIAL  E 

OMPENS 

0=   I 

~ 

300 

•  '     . 

DLTS 
1000 

uoo 

too 

roo 

GOO 
M) 
400 
300 
200 
100 
0 

~""~- 

\: 

~~~~-. 

-^. 

— 

— 

*-•, 

~—  ^. 

* 

V 

nr 

\ 

x 

//' 

"^ 

-^ 

//s 

\ 

\ 

"*x- 

^^ 

\ 

x 

2 

S 

/^ 

A 

1 

, 

s 

^- 

^ 

^T 

) 

^S 

^~ 

^^-* 

•*^=: 

— 

— 

— 

^ 

>^ 

/ 

^ 

^y 

*~^ 

-^. 

•*fc 

x 

f  

-" 

* 

^^ 

•^, 

^> 

-^ 

^ 

—1  — 

—  - 

_____ 

-.  — 

=rrT 

—- 

,  —  • 

01 

r?v 

T   IN 

RE 

iEIV 

x  c 

RC 

IT, 

<ILO 

.VAT 

TS 

30  W  50  CO  70  80 

Fig.  62.     Variation  of  Voltaqe  Transmission  Lines. 


100 


AL  TERNA  TING-CURRENT  PHENOMENA. 


RATIO  OF  RECEIVER  VOLTAGE  TO  SEN  DER  VOLTAGE:  a  =1.3 

INE  IMPEDANCE:  Z0=2.5.— ej" 


CONSTANT  GENERATOR  POTENTIAL  E0=IOOOl 


I.    ENERGY  CURRENT 

II.  "REACTIVE  CURRENT 

III.  TOTAL  CURRENT 

IV.  POTENTIAL  IN  NON-INDUCTIVE  RECEIVER  CIRCUIT  WITHOUT  COMPENSATION 


OUTPUT    N  RECEIVER  C  RCUIT,     KILOWATTS 


30  10  80  60  70  80  90 

Fig.  63.     Variation  of  Voltage  Transn\jssion  Lines. 

Energy  component  of  current,  gE,    (Curve  I.)  ; 

Reactive,  or  wattless  component  of  current,    bE,    (Curve  II.)  ; 
Total  current,  yE,    (Curve  III.)  ; 

for  the  following  conditions  : 

a  =  1.0  (Fig.  61)  ;     a  =    .7  (Fig.  62)  ;     a  =  1.3  (Fig.  63). 

For  the  non-inductive  receiver  circuit  (in  dotted  lines), 
the  curve  of  E.M.F.,  E,  and  of  the  current,  I  =  gE,  are 
added  in  the  three  diagrams  for  comparison,  as  Curves  IV. 
and  V. 

As  shown,  the  output  can  be  increased  greatly,  and  the 
potential  at  the  same  time  maintained  constant,  by  the  judi- 
cious use  of  shunted  reactance,  so  that  a  much  larger  out- 
put can  be  transmitted  over  the  line  at  no  drop,  or  even  at 
a  rise,  of  potential. 


RESISTANCE   OF   TRANSMISSION  LINES. 


101 


5.)    Maximum  Rise  of  Potential  at  Receiver  Circuit. 

70.  Since,  under  certain  circumstances,  the  potential  at 
the  receiver  circuit  may  be  higher  than  at  the  generator, 
it  is  of  interest  to  determine  what  is  the  maximum  value  of 
potential,  E,  that  can  be  produced  at  the  receiver  circuit 
with  a  given  generator  potential,  E0  . 

The  condition  is  that 


a  =  maxmum  or  —  =  mnmum : 
a2 


that  is, 


substituting, 


r0g  + 


(*0g  - 


and  expanding,  we  get, 

dg      =  °;      gss~"£'' 

—  a  value  which  is  impossible,  since  neither  r0  nor  g  can  be 
negative.  The  next  possible  value  is  g  —  0,  —  a  wattless 
circuit. 

Substituting  this  value,  we  get, 


and  by  substituting,  in 


, 

b  +  b0  =  0  ; 

that  is,  the  sum  of  the  susceptances  =  0,  or  the  condition 
of  resonance  is  present. 
Substituting, 

*=-*-£, 
we  have 


102  AL  TERNA  TING-CURRENT  PHENOMENA. 

The  current  in  this  case  is, 


or  the  same  as  if  the  line  resistance  were  short-circuited 
without  any  inductance. 

This  is  the  condition  of  perfect  resonance,  with  current 
and  E.M.F.  in  phase. 


\ 

s 

\ 

\ 

VOLT 

^ 

\ 

\ 

\ 

\ 

1SOO 
1700 
1COO 
1500 
-1400 

\ 

\ 

\\ 

\ 

\ 

CONSTANT  IMPRESSED  E.  M.  F.    Eo^lOOO 
"           LINE  IMPEDANCE  Z0=2.5-  € 
1    MAXIMUM  OUTPUT  BY  COMPENSATION 
II    MAXIMUM  EFFICIENCY  BY  COMPENSATIC 
III    NON-INDUCTiVE  RECEIVER  C  RCU  T 
IV    NON-INDUCTIVE  LINE  AND  NON-INDUCT 
RECEIVER  CIRCUIT 

If 

\ 

IN 

\ 

IVE 

1200 

1100 
1000 

l\ 

> 

s  1 

GOO 
800 
700 
COO 

* 

"-^ 

•^ 

SEC 

JFF 

C1EN_ 

*-. 

-* 

fa 

"N 

^ 

^ 

/^ 

5 

L 

•\ 

// 

tV 

i 

/ 

^ 

\ 

8 

/f 

// 

300 
200 
100 

A 

^ 

'       ., 

/\v 

^ 

tc)P 

^> 

4 

/ 

^  — 

*.ovJS 

^"\ 
PUT 

PUT 

K.W 

0        ' 

i)  i 

it 

Fig.  64.    Efficiency  and  Output  of  Transmission  Line. 

71.  As  summary  to  this  chapter,  in  Fig.  64  are  plotted, 
for  a  constant  generator  E.M.F.,  E0  =  1000  volts,  and  a 
line  impedance,  Z0  =  2.5  —  6/,  or,  r0  =  2.5  ohms,  x0  =  6 
ohms,  z0  =  6.5  ohms  ;  and  with  the  receiver  output  as 


RESISTANCE   OF  TRANSMISSION  LINES.  103 

abscissae   and    the    receiver  voltages    as    ordinates,   curves 
representing  — 

the  condition  of  maximum  output,  (Curve      I.)  ; 

the  condition  of  maximum  efficiency,  (Curve    II.)  ; 

the  condition  b  =  0,  or  a  non-inductive  receiver  cir- 
cuit, (Curve  III.)  ; 

the  condition  b  =  0,   b0  =  0,  or  a  non-inductive  line  and  non- 
inductive  receiver  circuit. 

In  conclusion,  it  may  be  remarked  here  that  of  the 
sources  of  susceptance,  or  reactance, 

a  choking  coil  or  reactive  coil  corresponds  to  an  inductance ; 
a  condenser  corresponds  to  a  condensance  ; 

a  polarization  cell  corresponds  to  a  condensance  ; 

a  synchronizing  alternator  (motor  or  generator)  corresponds  to 

an  inductance  or  a  condensance,  at  will; 
an  induction  motor  or  generator  corresponds  to  an  inductance. 

The  choking  coil  and  the  polarization  cell  are  specially 
suited  for  series  reactance,  and  the  condenser  and  syn- 
chronizer for  shunted  susceptance. 


104  ALTERNATING-CURRENT  PHENOMENA. 


CHAPTER    X. 

EFFECTIVE    RESISTANCE    AND    REACTANCE. 

72.    The  resistance  of  an  electric  circuit  is  determined :  — 

1.)  By  direct  comparison  with  a  known  resistance  (Wheat- 
stone  bridge  method,  etc.). 

This  method  gives  what  may  be  called  the  true  ohmic 
resistance  of  the  circuit. 

2.)    By  the  ratio  : 

Volts  consumed  in  circuit 

Amperes  in  circuit 

In  an  alternating-current  circuit,  this  method  gives,  not 
the  resistance  of  the  circuit,  but  the  impedance, 


3.)    By  the  ratio  : 

r__  Power  consumed  . 

(Current)2 

where,  however,  the  "power"  does  not  include  the  work 
done  by  the  circuit,  and  the  counter  E.M.Fs.  representing 
it,  as,  for  instance,  in  the  case  of  the  counter  E.M.F.  of  a 
motor. 

In  alternating-current  circuits,  this  value  of  resistance  is 
the  energy  coefficient  of  the  E.M.F., 

_  Energy  component  of  E.M.F. 

Total  current 

It  is  called  the  effective  resistance  of  the  circuit,  since  it 
represents  the  effect,  or  power,  expended  by  the  circuit. 
The  energy  coefficient  of  current, 

a._  Energy  component  of  current 

Total  E.M.F. 
is  called  the  effective  conductance  of  the  circuit. 


EFFECTIVE  RESISTANCE  AND   REACTANCE.        105 

In  the  same  way,  the  value, 

_  Wattless  component  of  E.M.F. 

Total  current 
is  the  effective  reactance,  and 

,  _  Wattless  component  of  current 
TotafE.M.F. 

is  the  effective  susceptance  of  the  circuit. 

While  the  true  ohmic  resistance  represents  the  expendi- 
ture of  energy  as  heat  inside  of  the  electric  conductor  by  a 
current  of  uniform  density,  the  effective  resistance  repre- 
sents the  total  expenditure  of  energy. 

Since,  in  an  alternating-current  circuit  in  general,  energy 
is  expended  not  only  in  the  conductor,  but  also  outside  of 
it,  through  hysteresis,  secondary  currents,  etc.,  the  effective 
resistance  frequently  differs  from  the  true  ohmic  resistance 
in  such  way  as  to  represent  a  larger  expenditure  of  energy. 

In  dealing  with  alternating-current  circuits,  it  is  necessary, 
therefore,  to  substitute  everywhere  the  values  "effective  re- 
sistance," "effective  reactance,"  "effective  conductance," 
and  "  effective  susceptance,"  to  make  the  calculation  appli- 
cable to  general  alternating-current  circuits,  such  as  induc- 
tances, containing  iron,  etc. 

While  the  true  ohmic  resistance  is  a  constant  of  the 
circuit,  depending  only  upon  the  temperature,  but  not  upon 
the  E.M.F.,  etc.,  the  effective  resistance  and  effective  re- 
actance are,  in  general,  not  constants,  but  depend  upon 
the  E.M.F.,  current,  etc.  This  dependence  is  the  cause 
of  most  of  the  difficulties  met  in  dealing  analytically  with 
alternating-current  circuits  containing  iron. 

73.  The  foremost  sources  of  energy  loss  in  alternating- 
current  circuits,  outside  of  the  true  ohmic  resistance  loss, 
are  as  follows : 

1.)    Molecular  friction,  as, 

a.)    Magnetic  hysteresis ; 
b.)   Dielectric  hysteresis. 


106  .ALTERNATING-CURRENT  PHENOMENA. 

2.)   Primary  electric  currents,  as, 

a.}   Leakage  or  escape  of  current  through  the  insu- 
lation, brush  discharge  ;  b.)  Eddy  currents  in 
the  conductor  or  unequal  current  distribution. 
3.)  Secondary  or  induced  currents,  as, 

a.)  Eddy  or  Foucault  currents  in  surrounding  mag- 
netic materials  ;  b.}  Eddy  or  Foucault  currents 
in  surrounding  conducting  materials  ;  c.}  Sec- 
ondary currents  of  mutual  inductance  in  neigh- 
boring circuits. 

4.)  Induced  electric  charges,  electrostatic  influence. 
While  all  these  losses  can  be  included  in  the  terms  effec- 
tive resistance,  etc.,  only  the  magnetic  hysteresis  and  the 
eddy  currents  in  the  iron  will  form  the  subject  of  what  fol- 
lows, since  they  are  the  most  frequent  and  important  sources 
of  energy  loss. 

Magnetic  Hysteresis. 

74.  In  an  alternating-current  circuit  surrounded  by  iron 
or  other  magnetic  material,  energy  is  expended  outside  of 
the  conductor  in  the  iron,  by  a  kind  of  molecular  friction, 
which,  when  the  energy  is  supplied  electrically,  appears  as 
magnetic  hysteresis,  and  is  caused  by  the  cyclic  reversals  of 
magnetic  flux  in  the  iron  in  the  alternating  magnetic  field. 

To  examine  this  phenomenon,  first  a  circuit  may  be  con- 
sidered, of  very  high  inductance,  but  negligible  true  ohmic 
resistance ;  that  is,  a  circuit  entirely  surrounded  by  iron,  as, 
for  instance,  the  primary  circuit  of  an  alternating-current 
transformer  with  open  secondary  circuit. 

The  wave  of  current  produces  in  the  iron  an  alternating 
magnetic  flux  which  induces  in  the  electric  circuit  an  E.M.F., 
—  the  counter  E.M.F.  of  self-induction.  If  the  ohmic  re- 
sistance is  negligible,  that  is,  practically  no  E.M.F.  con- 
sumed by  the  resistance,  all  the  impressed  E.M.F.  must  be 
consumed  by  the  counter  E.M.F.  of  self-induction,  that  is, 
the  counter  E.M.F.  equals  the  impressed  E.M.F.  ;  hence,  if 


EFFECTIVE   RESISTANCE   AND   REACTANCE. 


107 


the  impressed  E.M.F.  is  a  sine  wave,  the  counter  E.M.F., 
and,  therefore,  the  magnetic  flux  which  induces  the  counter 
E.M.F.  must  follow  a  sine  wave  also.  The  alternating  wave 
of  current  is  not  a  sine  wave  in  this  case,  but  is  distorted 
by  hysteresis.  It  is  possible,  however,  to  plot  the  current 
wave  in  this  case  from  the  hysteretic  cycle  of  magnetic  flux. 
From  the  number  of  turns,  n,  of  the  electric  circuit, 
the  effective  counter  E.M.F.,  E,  and  the  frequency,  N, 
of  the  current,  the  maximum  magnetic  flux,  <j>,  is  found 
by  the  formula  : 


hence, 


E  108 


A  maximum  flux,  <£,  and  magnetic  cross-section,  S,  give 
the  maximum  magnetic  induction,  (B  =  $  /  6". 

If  the  magnetic  induction  varies  periodically  between 
+  (B  and  —  (B,  the  M.M.F.  varies  between  the  correspond- 
ing values  -f  ff  and  —  JF,  and  describes  a  looped  curve,  the 
cycle  of  hysteresis. 

If  the  ordinates  are  given  in  lines  of  magnetic  force,  the 
abscissae  in  tens  of  ampere-turns,  then  the  area  of  the  loop 
equals  the  energy  consumed  by  hysteresis  in  ergs  per  cycle. 

From  the  hysteretic  loop  the  instantaneous  value  of 
M.M.F.  is  found,  corresponding  to  an  instantaneous  value 
of  magnetic  flux,  that  is,  of  induced  E.M.F.  ;  and  from  the 
M.M.F.,  JF,  in  ampere-turns  per  unit  length  of  magnetic  cir- 
cuit, the  length,  /,  of  the  magnetic  circuit,  and  the  number  of 
turns,  «,  of  the  electric  circuit,  are  found  the  instantaneous 
values  of  current,  i,  corresponding  to  a  M.M.F.,  JF;  that  is, 
magnetic  induction  (B,  and  thus  induced  E.M.F.  e,  as  : 


75.  In  Fig.  65,  four  magnetic  cycles  are  plotted,  with 
maximum  values  of  magnetic  inductions,  (B  =  2,000,  6,000, 
10,000,  and  16,000,  and  corresponding  maximum  M.M.Fs., 


108 


AL  TERNA  TING-CURRENT  PHENOMENA. 


SF  =  1.8,  2.8,  4.3,  20.0.  They  show  the  well-known  hys- 
teretic  loop,  which  becomes  pointed  when  magnetic  satu- 
ration is  approached. 

These  magnetic  cycles  correspond  to  average  good  sheet 
iron  or  sheet  steel,  having  a  hysteretic  coefficient,  77  =  .0033, 
and  are  given  with  ampere-turns  per  cm  as  abscissae,  and 
kilo-lines  of  magnetic  force  as  ordinates. 


a 


M 


«</.  65.    Hysteretic  Cycle  of  Sheet  Iron. 

In  Figs.  66,  67,  68,  and  69,  the  curve  of  magnetic  in- 
duction as  derived  from  the  induced  E.M.F.  is  a  sine  wave. 
For  the  different  values  of  magnetic  induction  of  this  sine 
curve,  the  corresponding  values  of  M.M.F.,  hence  of  current, 
are  taken  from  Fig.  65,  and  plotted,  giving  thus  the  exciting 
current  required  to  produce  the  sine  wave  of  magnetism ; 
that  is,  the  wave  of  current  which  a  sine  wave  of  impressed 
E.M.F.  will  send  through  the  circuit. 


EFFECTIVE  RESISTANCE  AND  REACTANCE.        109 

As  shown  in  Figs.  66,  67,  68,  and  69,  these  waves  of 
alternating  current  are  not  sine  waves,  but  are  distorted  by 
the  superposition  of  higher  harmonics,  and  are  complex 
harmonic  waves.  They  reach  their  maximum  value  at  the 
same  time  with  the  maximum  of  magnetism,  that  is,  90° 


1=2000 


1.6 


N 


^ 


\ 


(Bfeooo 


T2.8 


3  =2.S 


M\ 


\\ 


Figs.  66  and  67.     Distortion  of  Current  Waue  by  Hysteresis. 

ahead  of  the  maximum  induced  E.M.F.,  and  hence  about 
90°  behind  the  maximum  impressed  E.M.F.,  but  pass  the 
zero  line  considerably  ahead  of  the  zero  value  of  magnet- 
ism, or  42°,  52°,  50°,  and  41  °,  respectively. 

The  general  character  of  these  current  waves  is,  that  the 
maximum  point  of  the  wave  coincides  in  time  with  the  max- 


110 


ALTERNA  TING-CURRENT  PHENOMENA. 


imum  point  of  the  sine  wave  of  magnetism  ;  but  the  current 
wave  is  bulged  out  greatly  at  the  rising,  and  hollowed  in  at 
the  decreasing,  side.  With  increasing  magnetization,  the 
maximum  of  the  current  wave  becomes  more  pointed,  as 
shown  by  the  curve  of  Fig.  68,  for  (B  =  10,000  ;  and  at  still 


(B- 


10000 


4. 


& 


NX 


\L 


.  16000 


20 


\ 


G 


13 


\ 


F/SfS.  88  and  69.    Distortion  of  Current  Waue  by  Hysteresis. 

higher  saturation  a  peak  is  formed  at  the  maximum  point, 
as  in  the  curve  of  Fig.  69,  for  (B  =  16,000.  This  is  the  case 
when  the  curve  of  magnetization  reaches  within  the  range  of 
magnetic  saturation,  since  in  the  proximity  of  saturation  the 
current  near  the  maximum  point  of  magnetization  has  to 
rise  abnormally  to  cause  even  a  small  increase  of  magneti- 
zation. The  four  curves,  Figs.  66,  67,  68,  and  69,  are  not 
drawn  to  the  same  scale.  The  maximum  values  of  M.M.F., 


EFFECTIVE  RESISTANCE  A.\D  REACTANCE-     111 

corresponding  to  the  maximum  values  of  magnetic  induction, 
(B  =  2,000,  6,000,  10,000,  and  16,000  lines  of  force  per  cm2, 
'arc  &  =  1.8,  2.8,  4.3,  and  20.0  ampere-turns  per  cm.  In 
the  different  diagrams  these  are  represented  in  the  ratio  of 
8  :  6  :  4  :  1,  in  order  to  bring  the  current  curves  to  approxi- 
mately the  same  height.  The  M.M.F.,  in  C.G.S.  units,  is 
J#r=47r/103r  =  1.257  IF. 

76.  The  distortion  of  the  wave  of  magnetizing  current 
is  as  large  as  shown  here  only  in  an  iron-closed  magnetic 
circuit  expending  energy  by  hysteresis  only,  as  in  an  iron- 
clad transformer  on  Open  secondary  circuit.     As  soon  as  the 
circuit  expends  energy  in  any  other  way,  as  in  resistance,  or 
by  mutual  inductance,  or  if  an  air-gap  is  introduced  in  the 
magnetic  circuit,  the  distortion  of  the  current  wave  rapidly 
decreases  and  practically  disappears,  and  the  current  becomes 
more  sinusoidal.     That  is,  while  the  distorting  component 
remains  the  same,  the  sinusoidal  component  of  the  current 
greatly  increases,  and  obscures  the  distortion.     For  example, 
in  Figs.  70  and  71,  two  waves  are  shown,  corresponding  in 
magnetization  to  ^the  curve  of    Fig.  67,  as  the  one  most 
distorted.     The  curve  in  Fig.  70  is  the  current  wave  of  a 
transformer  at  TV  load.     At  higher  loads  the  distortion  is 
correspondingly  still  less,  except  where  the  magnetic  flux  of 
self-induction,  that  is,  flux  passing  between  primary  and  sec- 
ondary, and  increasing  proportionally  to  the  load,  is  so  large 
as  to  reach  saturation,  in  which  .case  a  distortion  appears 
again  and  increases  with  increasing  load.     The  curve  of  Fig. 
71  is  the  exciting  current  of  a  magnetic  circuit  containing 
an  air-gap  whose  length  equals  ?^  the  length  of  the  magnetic 
circuit.    These  two  curves  are  drawn  to  £  the  size  of  the  curve 
in  Fig.  67.    As  shown,  both  curves  are  practically  sine  waves. 
The  sine  curves  of  magnetic  flux  are  shown  dotted  as  <£. 

77.  The  distorted  wave  of  current  can  be  resolved  into 
two  components  :  A  true  sine  wave  of  equal  effective  intensity 
nnd  equal  power  to  the  distorted  wave,  called  the  equivalent 


112 


ALTERNATING-CURRENT  PHENOMENA. 


sine  wave,  and  a  wattless  JiigJier  harmonic,  consisting  chiefly 
of  a  term  of  triple  frequency. 

In  Figs.  66  to  71  are  shown,  as  /,  the  equivalent  sine' 


\ 


\ 


v 


\ 


Figs.  70  and  71.    Distortion  of  Current  Wave  by  Hysteresis. 

waves  and  as  i,  the  difference  between  the  equivalent  sine 
wave  and  the  real  distorted  wave,  which  consists  of  wattless 
complex  higher  harmonics.  The  equivalent  sine  wave  of 
M.M.F.  or  of  current,  in  Figs.  66  to  69,  leads  the  magnet- 


EFFECTIVE  RESISTANCE  AND  REACTANCE.        113 

ism  by  34°,  44°,  38°,  and  15°. 5,  respectively.  In  Fig.  71 
the  equivalent  sine  wave  almost  coincides  with  the  distorted 
curve,  and  leads  the  magnetism  by  only  9°. 

It  is  interesting  to  note,  that  even  in  the  greatly  dis- 
torted curves  of  Figs.  66  to  68,  the  maximum  value  of  the 
equivalent  sine  wave  is  nearly  the  same  as  the  maximum 
value  of  the  original  distorted  wave  of  M.M.F.,  so  long  as 
magnetic  saturation  is  not  approached,  being  1.8,  2.9,  and 
4.2,  respectively,  against  1.8,  2.8,  and  4.3,  the  maximum 
values  of  the  distorted  curve.  Since,  by  the  definition,  the 
effective  value  of  the  equivalent  sine  wave  is  the  same  as 
that  of  the  distorted  wave,  it  follows,  that  this  distorted 
wave  of  exciting  current  shares  with  the  sine  wave  the 
feature,  that  the  maximum  value  and  the  effective  value 
have  the  ratio  of  V2  -f-  1.  Hence,  below  saturation,  the 
maximum  value  of  the  distorted  curve  can  be  calculated 
from  the  effective  value  —  which  is  given  by  the  reading 
of  an  electro-dynamometer  —  by  using  the  same  ratio  that 
applies  to  a  true  sine  wave,  and  the  magnetic  characteris- 
tic can  thus  be  determined  by  means  of  alternating  cur- 
rents, with  sufficient  exactness,  by  the  electro-dynamometer 
method,  in  the  range  below  saturation. 

78.  In  Fig.  72  is  shown  the  true  magnetic  character- 
istic of  a  sample  of  good  average  sheet  iron,  as  found  by 
the  method  of  slow  reversals  with  the  magnetometer  ;  for 
comparison  there  is  shown  in  dotted  lines  the  same  char- 
acteristic, as  determined  with  alternating  currents  by  the 
electro-dynamometer,  with  ampere-turns  per  cm  as  ordi- 
nates,  and  magnetic  inductions  as  abscissas.  As  repre- 
sented, the  two  curves  practically  coincide  up  to  a  value  of 
&  =  13,000  ;  that  is,  up  to  the  highest  inductions  practicable 
in  alternating-current  apparatus.  For  higher  saturations, 
the  curves  rapidly  diverge,  and  the  electro-dynamometer 
curve  shows  comparatively  small  M.M.Fs.  producing  appar- 
ently very  high  magnetizations. 


114 


AL  TERN  A  TING-CUR  RE  KT  PHENOMENA. 


The  same  Fig.  72  gives  the  curve  of  hysteretic  loss,  in 
ergs  per  cm3  and  cycle,  as  ordinates,  and  magnetic  induc- 
tions as  abscissae. 


TT 

\ 

/ 

/ 

/       / 
/ 

18 

/ 

/ 
/ 

17 

r 

1 
1 

/I 

' 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

' 

/ 

/ 

1 

/ 

/ 

I. 

' 

/ 

// 

/ 

/ 

1 

/ 

I 

/ 

/ 

/ 

/' 

/ 

^ 

^ 

/ 

^" 

^ 

^^ 

^ 

;  j^ 

* 

x 

2=EE 

x^^ 

£=1,000  2,000   3,000  1.0CO  5,000  6,000  7,000  8,000   9,000  10,000  11,000  12,000  13,0<W14,000  15, 
Fig.  72.    Magnetization  and  Hysteresis  Curve. 

woie.ooo  iv.ow 

The  electro-dynamometer  method  of  determining  the 
magnetic  characteristic  is  preferable  for  use  with  alter- 
nating-current apparatus,  since  it  is  not  affected  by  the 
phenomenon  of  magnetic  "creeping,"  which,  especially  at 


EFFECTIVE  RESISTANCE  AND  REACTANCE.         115 

low  densities,  may  in  the  magnetometer  tests  bring  the  mag- 
netism very  much  higher,  or  the  M.M.F.  lower,  than  found 
in  practice  in  alternating-current  apparatus. 

So  far  as  current  strength"  and  energy  consumption  are 
concerned,  the  distorted  wave  can  be  replaced  by  the  equi- 
valent sine  wave,  and  the  higher  harmonics  neglected. 

All  the  measurements  of  alternating  currents,  with  the 
single  exception  of  instantaneous  readings,  yield  the  equiv- 
alent sine  wave  only,  and  suppress  the  higher  harmonic  ; 
since  all  measuring  instruments  give  either  the  mean  square 
of  the  current  wave,  or  the  mean  product  of  instantaneous 
values  of  current  and  E.M.F.,  which,  by  definition,  are  the 
same  in  the  equivalent  sine  wave  as  in  the  distorted  wave. 

Hence,  in  all  practical  applications,  it  is  permissible  to 
neglect  the  higher  harmonic  altogether,  and  replace  the  dis- 
torted wave  by  its  equivalent  sine  wave,  keeping  in  mind, 
however,  the  existence  of  a  higher  harmonic  as  a  possible 
disturbing  factor  which  may  become  noticeable  in  those  cases 
where  the  frequency  of  the  higher  harmonic  is  near  the  fre- 
quency of  resonance  of  the  circuit,  that  is,  in  circuits  con- 
taining capacity  besides  the  inductance. 

79.  The  equivalent  sine  wave  of  exciting  current  leads 
the  sine  wave  of  magnetism  by  an  angle  a,  which  is  called 
the  angle  of  Jiysteretic  advance  of  phase.  Hence  the  cur- 
rent lags  behind  the  E.M.F  by  ^  90°  —  a,  and  the  power 


is  therefore,     p=f£  cog  (9QO  _  a)  =  /E  sin  a 

Thus  the  exciting  current,  7,  consists  of  an  energy  compo- 
nent, /  sin  a,  called  the  Jiysteretic  or  magnetic  energy  current, 
and  a  wattless  component,  /  cos  a,  which  is  called  the  mag- 
netizing current.  Or,  conversely,  the  E.M.F.  consists  of  an 
energy  component,  E  sin  a,  the  Jiysteretic  energy  E.M.F., 
and  a  wattless  component,  E  cos  a,  the  E.M.F.  of  self- 
induction. 

Denoting  the  absolute  value  of  the  impedance  of  the 


116  A  L  TERNA  TING-CURRENT  PHENOMENA  . 

circuit,  E  1  1,  by  s,  —  where  s  is  determined  by  the  mag- 
netic characteristic  of  the  iron,  and  the  shape  of  the 
magnetic  and  electric  circuits,  —  the  impedance  is  repre- 
sented, in  phase  and  intensity,  by  the  symbolic  expression, 

Z  =  r  —  jx  =  z  sin  a  —  jz  cos  a  ; 
and  the  admittance  by, 

Y  =  g  +  j  b  =  -  sin  a  -j-  j  -  cos  a  =  y  sin  a  -f-  jy  cos  a. 

z  z 

The  quantities,  z,  r,  x,  and  y,  g,  b,  are,  however,  not 
constants  as  in  the  case  of  the  circuit  without  iron,  but 
depend  upon  the  intensity  of  magnetization,  (B,  —  that  is, 
upon  the  E.M.F.  This  dependence  complicates  the  investi- 
gation of  circuits  containing  iron. 

In  a  circuit  entirely  inclosed  by  iron,  a  is  quite  consider- 
able, ranging  from  30°  to  50°  for  values  below  saturation. 
Hence,  even  with  negligible  true  ohmic  resistance,  no  great 
lag  can  be  produced  in  ironclad  alternating-current  circuits. 

80.  The  loss  of  energy  by  hysteresis  due  to  molecular 
friction  is,  with  sufficient  exactness,  proportional  to  the 
1.6th  power  of  magnetic  induction  <&.  Hence  it  can  be  ex- 
pressed by  the  formula  : 


where  — 

IV  a  =  loss  of  energy  per  cycle,  in  ergs  or  (C.G.S.)  units  (=  10~7 
Joules)  per  cm8, 

(ft  =  maximum  magnetic  induction,  in  lines  of  force  per  cm2,  and 

77  =  the  coefficient  of  hysteresis. 

This  I  found  to  vary  in  iron  from  .00124  to  .0055.  As  a 
fair  mean,  .0033  *  can  be  accepted  for  good  average  annealed 
sheet  iron  or  sheet  steel.  In  gray  cast  iron,  17  averages 
.013  ;  it  varies  from  .0032  to  .028  in  cast  steel,  according 
to  the  chemical  or  physical  constitution  ;  and  reaches  values 
as  high  as  .08  in  hardened  steel  (tungsten  and  manganese 

*  At  present,  with  the  improvements  in  the  production  and  selection  of  sheet  steel  far 
alternating  apparatus,  .0025  can  be  considered  a  fair  average  in  selected  material  (1899). 


EFFECTIVE  RESISTANCE  AND  REACTANCE.       117 

steel).  Soft  nickel  and  cobalt  have  about  the  same  co- 
efficient of  hysteresis  as  gray  cast  iron ;  in  magnetite  I 
found  rj  =  .023. 

In  the  curves  of  Fig.  62  to  69,  r,  =  .0033. 

At  the  frequency,  N,  the  loss  of  power  in  the  volume,  V, 
is,  by  this  formula,  — 

P=-t]N  F&1-6 10  - '  watts 


where  S  is  the  cross-section  of  the  total  magnetic  flux,  <£. 

The    maximum    magnetic    flux,    <E>,    depends    upon    the 
counter  E.M.F.  of  self-induction, 

E  =  V2  -IT  Nn  4>  10  - 8, 


V2  TT  Nn 

where  n  =  number  of  turns  of  the  electric  circuit. 

Substituting  this    in   the   value  of   the  power,   P,    and 
canceling,  we  get,  — 

E1-'          FIO  5-8  E™     F108 


no5-8        Ka     no3 
»  where  ^  =  ^  o.R  i.«  oi.fi  ..,..  =  58  -n 


T/- 

or,  substituting  •>;  =  .0033,  we  have  ^4  =  191.4  —^ — —  ; 

o    '    /?  * 

or,  substituting  F=  SL,  where  L  =  length  of  magnetic  circuit, 
•n  L  10 5-8  58 » Z 103  Z 

— — 


and  103       191.4  E 


In  Figs.  73,  74,  and  75,  is  shown  a  curve  of  hysteretic 
loss,  with  the  loss  of  power  as  ordinates,  and 

in  curve  73,  with  the  E.M.F.,  E,  as  abscissae,  for  L  =  6, 
S  =  20,  N=  100,  and  n  =  100  ; 


118 


AL  TERNA  TING-CURRENT  PHENOMENA. 


RELATION 

BE 

TW  = 

EN 

EA 

NDP 

F 

OR 

_— 

5,8 

=  20 

N  = 

10 

r5 

=  1 

oo 

/ 

/ 

/ 

K 

/ 

o 

/ 

^/ 

Q. 

x 

^ 

x 

X 

^ 

x 

x 

x 

x 

^ 

X* 

^ 

.  • 



^ 

E.IV 

l.F. 

Fig.  73.    Hysteresis  Loss  as  Function  of  £.  M.  F. 


BETW 
OR  LT6.  S=20,  ^ 


=  100.E= 


SO  100  160  200  250  300 

Fig.  74.    Hysteresis  Loss  as  Function  of  Number  of  Turns. 


EFFECTIVE   RESISTANCE   AND   REACTANCE. 


119 


II    I    I    II    I 


RELATION  BETWEEN   N  AND  P 
FOR  8=20,  L=6, 71  =  100.  E  =  100. 


Fig.  75.     Hysteresis  Loss  as  Function  of  Cycles. 

in  curve  74,  with  the  number  of  turns  as  abscissae,  for 
Z  =  6,  S  =  20,  JV=  100,  and  E  =  100 ; 

in  curve  75,  with  the  frequency,  JV,  or  the  cross-section,  S, 
as  abscissae,  for  L  =  6,  n  =  100,  and  E  =  100. 

As  shown,  the  hysteretic  loss  is  proportional  to  the  1.6th 
power  of  the  E.M.F.,  inversely  proportional  to  the  1.6th 
power  of  the  number  of  turns,  and  inversely  proportional  to 
the  .6th  power  of  frequency,  and  of  cross-section. 

81.  If  g  =  effective  conductance,  the  energy  compo- 
nent of  a  current  is  /  =  Eg,  and  the  energy  consumed  in 
a  conductance,  g,  is  P  =  IE  =  Ezg. 

Since,  however  : 

P  =  A ,  we  have  A =  E2  g ; 

or 

A  58r)L  10s 


191.4 


From  this  we  have  the  following  deduction : 


120 


ALTERNA TING-CURRENT  PHENOMENA. 


The  effective  conductance  due  to  magnetic  hysteresis  is 
proportional  to  the  coefficient  of  hysteresis,  rj,  and  to  the  length 
of  the  magnetic  circuit,  L,  and  inversely  proportional  to  the 
Jj!h  power  of  the  E.M.F.,  to  the  .6th  power  of  the  frequency, 
N,  and  of  the  cross-section  of  tlie  magnetic  circuit,  S,  and  to 
tlie  1.6th  power  of  the  number  of  turns,  n. 

Hence,  the  effective  hysteretic  conductance  increases 
with  decreasing  E.M.F.,  and  decreases  with  increasing 


RELATION 
FOR  L=6, 

BE- 

PWEEN     0AND  E 
00.  S  =  20,?l  =  1O 

V 

\ 

\ 

\ 

^ 

\ 

> 

^. 

.^^ 

__9 

a 

1  -, 

-  —  -. 

—  ^ 

——  . 

. 

•  , 

E 

Ftg.  76.    Hysteresis  Conductance  as  Function  of  E.M.F. 

E.M.F. ;  it  varies,  however,  much  slower  than  the  E.M.F., 
so  that,  if  the  hysteretic  conductance  represents  only  a  part 
of  the  total  energy  consumption,  it  can,  within  a  limited 
range  of  variation  —  as,  for  instance,  in  constant  potential 
transformers  —  be  assumed  as  constant  without  serious 
error. 

In  Figs.  76,  77,  and  78,  the  hysteretic  conductance,  g,  is 
plotted,  for  L  =  6,  E  =  100,  N=  100,  5  =  20  and  n  =  100, 
respectively,  with  the  conductance,  g,  as  ordinates,  and  with 


EFFECTIVE  RESISTANCE  AND   REACTANCE. 


1-21 


RELATION  BETWEEN    Q  AND  N 
FOR  L-6,  E  =  IOO.  S  =  20,  n=IOO 


Fig.  77.     Hysteresis  Conductance  as  Function  of  Cycles, 


• 

R 

LAI 

,0, 

BE 

WE 

EN 

,AS 

D(/ 

FOP 

L= 

6,E 

=  1( 

50, 

00 

,8= 

2a 

\ 

b 

V 

a 

\ 

\ 

s 

\ 

X. 

E 

- 

T 

-NL 

\. 

M~B~ 

:RO 

•  —  , 

F  T 



r= 

200  250  300  350 

Fig.  78.    Hysteresis  Conductance  as  Function  of  Number  of  Turns. 


122  ALTERNATING-CURRENT  PHENOMENA. 

E  as  abscissae  in  Curve  76. 
.A^  as  abscissas  in  Curve  77. 
n  as  abscissas  in  Curve  78. 

As  shown,  a  variation  in  the  E.M.F.  of  50  per  cent 
causes  a  variation  in  g  of  only  14  per  cent,  while  a  varia- 
tion in  N  or  6"  by  50  per  cent  causes  a  variation  in  g  of  21 
per  cent. 

If  (R  =  magnetic  reluctance  of  a  circuit,  £FA  =  maximum 
M.M.F.,  I  —  effective  current,  since  /V2  =  maximum  cur- 
rent, the  magnetic  flux, 


(R  (R 

Substituting  this  in  the  equation  of  the  counter  E.M.F.  of 
self-induction 


we  have 


(R 
hence,  the  absolute  admittance  of  the  circuit  is 

(RIO8   =  a& 

E  ~  2  TT  n*N  ~  N  ' 

108 

where  a  =  ,  a  constant. 

2  TT  n 

Therefore,  the  absolute  admittance,  y,  of  a  circuit  of  neg- 
ligible resistance  is  proportional  to  the  magnetic  reluctance,  (R, 
and  inversely  proportional  to  the  frequency,  N,  and  to  the 
square  of  the  number  of  turns,  n. 

82.  In  a  circuit  containing  iron,  the  reluctance,  (R,  varies 
with  the  magnetization  ;  that  is,  with  the  E.M.F.  Hence 
the  admittance  of  such  a  circuit  is  not  a  constant,  but  is 
also  variable. 

In  an  ironclad  electric  circuit,  —  that  is,  a  circuit  whose 
magnetic  field  exists  entirely  within  iron,  such  as  the  mag- 
netic circuit  of  a  well-designed  alternating-current  trans- 


EFFECTIVE   RESISTANCE   AND   REACl^ANCE.        123 

former,  —  (R  is  the  reluctance  of  the  iron  circuit.     Hence, 
if  p.  =  permeability,  since  — 


and  g:A  =  jr/7=Zge  =  M.M.F., 


and  <R,       10L 


magnetic  flux, 


substituting  this  value  in  the  equation  of  the  admittance, 

(R  108  Z 109  z 

y=  -z-      nrv>  we  have  5— ; 


where  „       L  W         127Z10' 


TJierefore,  in  an  ironclad  circuit,  the  absolute  admittance, 
y,  is  inversely  proportional  to  the  frequency,  N,  to  the  perme- 
ability, JJL,  to  the  cross-section,  S,  and  to  the  square  of  the 
number  of  turns,  n  ;  and  directly  proportional  to  the  length 
of  the  magnetic  circuit,  L. 


The  conductance  is 


= 

and  the  admittance,  y  =  -  ; 

yv/u. 

hence,  the  angle  of  hysteretic  advance  is 


or,  substituting  for  A  and  z  (p.  117), 
NA         «Z1068 


or,  substituting 
J£ 

we  have        sin  a  =  — 

-4   ' 


1  24  AL  TERN  A  TING-CURRENT  PHENOMENA. 

which  is  independent  of  frequency,  number  of  turns,  and 
shape  and  size  of  the  magnetic  and  electric  circuit. 

Therefore,  in  an  ironclad  inductance,  tJie  angle  of  Jiysteretic 
advance,  a,  depends  upon  the  magnetic  constants,  permeability 
and  coefficient  of  hysteresis,  and  tipon  the  maximum  magnetic 
induction,  but  is  entirely  independent  of  the  frequency,  of  the 
shape  and  other  conditions  of  the  magnetic  and  electric  circuit  ; 
and,  therefore,  all  ironclad  'magnetic  circuits  constructed  of  the 
same  quality  of  iron  and  using  the  same  magnetic  density, 
give  the  same  angle  of  Jiysteretic  advance. 

The  angle  of  Jiysteretic  advance,  a,  in  a  closed  circuit 
transformer,  depends  tipon  tJie  quality  of  the  iron,  and  upon 
the  magnetic  density  only. 

The  sine  of  tJie  angle  of  Jiysteretic  advance  equals  4  times 
the  product  of  the  permeability  and  coefficient  of  hysteresis, 
divided  by  the  .4th  power  of  tJie  magnetic  density. 

83.  If  the  magnetic  circuit  is  not  entirely  ironclad, 
and  the  magnetic  structure  contains  air-gaps,  the  total  re- 
luctance is  the  sum  of  the  iron  reluctance  and  of  the  air 
reluctance,  or 

<R  =  (R  {  _|_  <Rfl  ; 

hence  the  admittance  is 


TJierefore,  in  a  circuit  containing  iron,  the  admittance  ts 
the  sum  of  the  admittance  due  to  the  iron  part  of  tJie  circuit, 
yi  =  a&i/  N,  and  of  the  admittance  due  to  the  air  part  of  the 
circuit,  ya  =  a  (&a  /  N,  if  the  iron  and  the  air  are  in  series  in 
the  magnetic  circuit. 

The  conductance,  g,  represents  the  loss  of  energy  in 
the  iron,  and,  since  air  has  no  magnetic  hysteresis,  is  not 
changed  by  the  introduction  of  an  air-gap.  Hence  the 
angle  of  hysteretic  advance  of  phase  is 


sm  a  =  — 

y 


EFFECTIVE   RESISTANCE   AND  REACTANCE.         125 

and  a  maximum, gjyt,  for  the  ironclad  circuit,  but  decreases 
with  increasing  width  of  the  air-gap.  The  introduction  of 
the  air-gap  of  reluctance,  (R0,  decreases  sin  a  in  the  ratio, 

<Rj 

«*  +  <*« ' 

In  the  range  of  practical  application,  from  (B  =  2,000  to 
(B  =  12,000,  the  permeability  of  iron  varies  between  900 
and  2,000  approximately,  while  sin  a  in  an  ironclad  circuit 
varies  in  this  range  from  .51  to  .69.  In  air,  /t  =  1. 

If,  consequently,  one  per  cent  of  the  length  of  the  iron 
consists  of  an  air-gap,  the  total  reluctance  only  varies  through 
the  above  range  of  densities  in  the  proportion  of  1^  to  Ig^, 
or  about  6  per  cent,  that  is,  remains  practically  constant ; 
while  the  angle  of  hysteretic  advance  varies  from  sin  a  =  .035 
to  sin  a  =  .064.  Thus  g  is  negligible  compared  with  b,  and 
b  is  practically  equal  to  j. 

Therefore,  in  an  electric  circuit  containing  iron,  but 
forming  an  open  magnetic  circuit  whose  air-gap  is  not  less 
than  T^  the  length  of  the  iron,  the  susceptance  is  practi- 
cally constant  and  equal  to  the  admittance,  so  long  as 
saturation  is  not  yet  approached,  or, 

b  =  <Ra  /  N,  or  :  x  =  N/  (Ra. 

The  angle  of  hysteretic  advance  is  small,  below  4°,  and  the 
hysteretic  conductance  is, 

-=       A 

EAN*  ' 

The  current  wave  is  practically  a  sine  wave. 

As  an  instance,  in  Fig.  71,  Curve  II.,  the  current  curve 
of  a  circuit  is  shown,  containing  an  air-gap  of  only  ^  of 
the  length  of  the  iron,  giving  a  current  wave  much  resem- 
bling the  sine  shape,  with  an  hysteretic  advance  of  9°. 

84.    To    determine  the  electric  constants   of    a  circuit 
containing  iron,  we  shall  proceed  in  the  following  way : 
Let  — 

E  =  counter  E.M.F.  of  self-induction  ; 


126  ALTERNATING-CURRENT  PHENOMENA. 

then  from  the  equation, 
E  = 


where, 

N  '=  frequency, 

n  =  number  of  turns, 


we  get  the  magnetism,  <£,  and  by  means  of  the  magnetic  cross 
section,  S,  the  maximum  magnetic  induction  :  ($>  =  ®  /  S. 

From  (B,  we  get,  by  means  of  the  magnetic  characteristic 
of  the  iron,  the  M.M.F.,  =  F  ampere-turns  per  cm  length, 
where 


if  OC  =  M.M.F.  in  C.G.S.  units. 

Hence, 
if   Z,  =  length    of   iron    circuit,  JFj  =  Z,  F  =  ampere-turns    re- 

quired in  the  iron  ; 
if  La  =  length  of  air  circuit,  CFa  =  —  —  -  —  =  ampere-turns  re- 

quired in  the  air  ; 

hence,  CF=  JF,  -)-  $Fa  =  total  ampere  -turns,  maximum  value, 
and  JF/  V2  =  effective  value.     The  exciting  current  is 


and  the  absolute  admittance, 


If  SF,  is  not  negligible  as  compared  with  JFa,  this  admit- 
tance,^, is  variable  with  the  E.M.F.,  E. 

If  — 

V  =  volume  of  iron, 

rj   =  coefficient  of  hysteresis, 

the  loss  of  energy  by  hysteresis  due  to  molecular  magnetic 
friction  is, 


hence  the  hysteretic  conductance  is  g  =  lV/£?,  and  vari- 
able with  the  E.M.F.,  E. 


EFFECTIVE  RESISTANCE  AND  REACTANCE.        127 

The  angle  of  hysteretic  advance  is,  — 

sin  a=g/y; 

the  susceptance,  b  =  Vj*2  —  gz\ 

the  effective  resistance,  r  =  g  /  y*\ 

and  the  reactance,  x  =  b  / y*. 

85.  As  conclusions,  we  derive  from  this  chapter  the 
following :  — 

1.)  In  an  alternating-current  circuit  surrounded  by  iron, 
the  current  produced  by  a  sine  wave  of  E.M.F.  is  not  a  true 
sine  wave,  but  is  distorted  by  hysteresis,  and  inversely,  a 
sine  wave  of  current  requires  waves  of  magnetism  and 
E.M.F.  differing  from  sine  shape. 

2.)  This  distortion  is  excessive  only  with  a  closed  mag- 
netic circuit  transferring  no  energy  into  a  secondary  circuit 
by  mutual  inductance. 

3.)  The  distorted  wave  of  current  can  be  replaced  by 
the  equivalent  sine  wave  —  that  is  a  sine  wave  of  equal  effec- 
tive intensity  and  equal  power — and  the  superposed  higher 
harmonic,  consisting  mainly  of  a  term  of  triple  frequency, 
may  be  neglected  except  in  resonating  circuits. 

4.)  Below  saturation,  the  distorted  curve  of  current  and 
its  equivalent  sine  wave  have  approximately  the  same  max- 
imum value. 

5.)  The  angle  of  hysteretic  advance, — that  is,  the  phase 
difference  between  the  magnetic  flux  and  equivalent  sine 
wave  of  M.M.F.,  —  is  a  maximum  for  the  closed  magnetic 
circuit,  and  depends  there  only  upon  the  magnetic  constants 
of  the  iron,  upon  the  permeability,  yu.,  the  coefficient  of  hys- 
teresis, rj,  and  the  maximum  magnetic  induction,  as  shown*  in 

the  equation,  4 

sin  a  =  — f—i . 

&'4 

6.)  The  effect  of  hysteresis  can  be  represented  by  an 
admittance,  Y  —  g  +  j  b,  or  an  impedance,  Z  =  r  —  j x. 

7.)  The  hysteretic  admittance,  or  impedance,  varies  with 
the  magnetic  induction;  that  is,  with  the  E.M.F.,  etc. 


128  ALTERNATING-CURRENT  PHENOMENA. 

8.)  The  hysteretic  conductance,  £•,  is  proportional  to  the 
coefficient  of  hysteresis,  17,  and  to  the  length  of  the  magnetic- 
circuit,  L,  inversely  proportional  to  the  .4th  power  of  the 
E.M.F.,  E,  to  the  .6^h  power  of  frequency,  N,  and  of  the 
cross-section  of  the  magnetic  circuit,  S,  and  to  the  1.6th 
power  of  the  number  of  turns  of  the  electric  circuit,  ;/,  as 
expressed  in  the  equation, 

58  7  Z  103 


9.)    The  absolute  value  of  hysteretic  admittance,  — 


is  proportional  to  the  magnetic  reluctance  :  (R  =  (R,  -f  (Ra  , 
and  inversely  proportional  to  the  frequency,  N,  and  to  the 
square  of  the  number  of  turns,  n,  as  expressed  in  the 


>  _(«.  +  «„)  10- 

2-irNn* 

10.)    In  an  ironclad  circuit,  the  absolute  value  of  admit- 
tance is  proportional  to  the  length  of  the  magnetic  circuit, 
and  inversely  proportional  to  cross-section,  S,  frequency,  Ny 
permeability,  /*,  and  square  of  the  number  of  turns,  n,  or 
127  L  106 


11.)  In  an  open  magnetic  circuit,  the  conductance,  gt  is 
the  same  as  in  a  closed  magnetic  circuit  of  the  same  iron  part. 

12.)  In  an  open  magnetic  circuit,  the  admittance,  yt  is 
practically  constant,  if  the  length  of  the  air-gap  is  at  least 
TJC  of  the  length  of  the  magnetic  circuit,  and  saturation  be 
not  approached. 

13.)  In  a  closed  magnetic  circuit,  conductance,  suscep- 
tance,  and  admittance  can  be  assumed  as  constant  through 
a  limited  range  only. 

14.)  From  the  shape  and  the  dimensions  of  the  circuits, 
and  the  magnetic  constants  of  the  iron,  all  the  electric  con- 
stants, gy  b,y;  r,  x,  z,  can  be  calculated. 


FOUCAULT  OR   EDDY  CURRENTS.  129 


CHAPTER    XI. 

FOUCAULT  OR  EDDY  CURRENTS. 

86.  While  magnetic  hysteresis  or  molecular  friction  is 
a  magnetic  phenomenon,  eddy  currents  are  rather  an  elec- 
trical phenomenon.  When  iron  passes  through  a  magnetic 
field,  a  loss  of  energy  is  caused  by  hysteresis,  which  loss, 
however,  does  not  react  magnetically  upon  the  field.  When 
cutting  an  electric  conductor,  the  magnetic  field  induces  a 
current  therein.  The  M.M.F.  of  this  current  reacts  upon 
and  affects  the  magnetic  field,  more  or  less  ;  consequently, 
an  alternating  magnetic  field  cannot  penetrate  deeply  into  a 
solid  conductor,  but  a  kind  of  screening  effect  is  produced, 
which  makes  solid  masses  of  iron  unsuitable  for  alternating 
fields,  and  necessitates  the  use  of  laminated  iron  or  iron 
wire  as  the  carrier  of  magnetic  flux. 

Eddy  currents  are  true  electric  currents,  though  flowing 
in  minute  circuits;  and  they  follow  all  the  laws  of  electric 
circuits. 

Their  E.M.F.  is  proportional  to  the  intensity  of  magneti- 
zation, (B,  and  to  the  frequency,  N. 

Eddy  currents  are  thus  proportional  to  the  magnetization, 
(B,  the  frequency,  N,  and  to  the  electric  conductivity,  y,  of 
the  iron  ;  hence,  can  be  expressed  by 


The  power  consumed  by  eddy  currents  is  proportional  to 
their  square,  and  inversely  proportional  to  the  electric  con- 
ductivity, and  can  be  expressed  by 

W= 


130  ALTERNATING-CURRENT  PHENOMENA. 

or,  since,  ($>N  is  proportional  to  the  induced  E.M.F.,  E,  in 
the  equation 


it  follows  that,  TJie  loss  of  power  by  eddy  currents  is  propor- 
tional to  the  square  of  the  E.M.F.,  and  proportional  to  tlie 
electric  conductivity  of  the  iron  ;  or, 

W=aE*y. 

Hence,   that    component   of    the    effective    conductance 
which  is  due  to  eddy  currents,  is 


that  is,  The  equivalent  conductance  due  to  eddy  currents  in 
the  iron  is  a  constant  of  the  magnetic  circuit  ;  it  is  indepen- 
dent of  ^M..^.,  frequency,  etc.,  but  proportional  to  the  electric 
conductivity  of  the  iron,  y. 

87.  Eddy  currents,  like  magnetic  hysteresis,  cause  an 
advance  of  phase  of  the  current  by  an  angle  of  advance,  ft  ; 
but,  unlike  hysteresis,  eddy  currents  in  general  do  not  dis- 
tort the  current  wave. 

The  angle  of  advance  of  phase  due  to  eddy  currents  is, 

sin/3  =  £, 

where  y  =  absolute  admittance  of  the  circuit,  g  =  eddy 
current  conductance. 

While  the  equivalent  conductance,  g,  due  to  eddy  cur- 
rents, is  a  constant  of  the  circuit,  and  independent  of 
E.M.F.,  frequency,  etc.,  the  loss  of  power  by  eddy  currents 
is  proportional  to  the  square  of  the  E.M.F.  of  self-induction, 
and  therefore  proportional  to  the  square  of  the  frequency 
and  to  the  square  of  the  magnetization. 

Only  the  energy  component,  g  E,  of  eddy  currents,  is  of 
interest,  since  the  wattless  component  is  identical  with  the 
wattless  component  of  hysteresis,  discussed  in  a  preceding 
chapter. 


FOUCAULT  OR  EDDY  CURRENTS. 


131 


88.    To  calculate  the  loss  of  power  by  eddy  currents  — 

Let  V  =  volume  of  iron  ; 

(B  =  maximum  magnetic  induction  ; 
N=  frequency; 

y    =  electric  conductivity  of  iron  ; 
£    =  coefficient  of  eddy  currents. 

The  loss  of  energy  per  cm3,  in  ergs  per  cycle,  is 


hence,  the  total  loss  of  power  by  eddy  currents  is 

W  =  e  y  VN*  (B2  10  -  7  watts, 
and  the  equivalent  conductance  due  to  eddy  currents  is 


o_   W  _   IQey/  __  .507ey/ 

£>               Tf"2           O      2    C^/2                   C«2          * 

where  : 

/  =  length  of  magnetic  circuit, 

d 

S  —  section  of  magnetic  circuit, 
n  =  number  of  turns  of  electric  circuit. 

The  coefficient  of  eddy  currents,  e, 
depends  merely  upon  the  shape  of  the 
constituent   parts  of  the  magnetic  cir- 
cuit ;    that   is,  whether  of  iron  plates 
or  wire,  and  the  thickness  of  plates  or 
the  diameter  of  wire,  etc. 

x    i  JC 

The  two  most  important  cases  are  : 

(a).    Laminated  iron. 
(b).    Iron  wire. 

1 

'  1 

89.    (a).    Laminated  Iron. 
Let,  in  Fig.  79, 

i 

d   =  thickness  of  the  iron  plates  ; 
(B  =  maximum  magnetic  induction  ; 
JV  =  frequency  ; 
y    =  electric  conductivity  of  the  iron. 

Fi 

1.79. 

132  ALTERNATING-CURRENT  PHENOMENA. 

Then,  if  x  is  the  distance  of  a  zone,  d  x,  from  the  center 
of  the  sheet,  the  conductance  of  a  zone  of  thickness,  */x, 
and  of  one  cm  length  and  width  is  y^x  ;  and  the  magnetic 
flux  cut  by  this  zone  is  (Bx.  Hence,  the  E.M.F.  induced  in 
this  zone  is 

8  E  =  V2  TrN($>  x,  in  C.G.S.  units. 

This  E.M.F.  produces  the  current : 

///=SJ£y</x  =  V2  TrN<$>  y  x  d  x,  in  C.G.S.  units, 

provided  the  thickness  of  the  plate  is  negligible  as  compared 
with  the  length,  in  order  that  the  current  may  be  assumed 
as  flowing  parallel  to  the  sheet,  and  in  opposite  directions 
on  opposite  sides  of  the  sheet. 

The  power  consumed  by  the  induced  current  in  this 
zone,  dx,  is 

dP  =  §EdI=  2 7T2^2(B2  y  x  Vx,  in  C.G.S.  units  or  ergs  per  second, 

and,  consequently,  the  total  power  consumed  in  one  cm2  of 
the  sheet  of  thickness,  d,  is 


=    C+*  dP  =  27rW2(B2y   C 


°  in  C.G.S.  units; 


the  power  consumed  per  cm3  of  iron  is,  therefore, 


. 

/  =  —  =  -  —  '-  —  ,  m  C.G.S.  units  or  erg-seconds, 
and  the  energy  consumed  per  cycle  and  per  cm3  of  iron  is 


N  6 

The  coefficient  of  eddy  currents  for  laminated  iron  is, 
therefore, 

c  =  ^-  =  1.645  d\ 


FOUCAULT  OR  EDDY  CURRENTS.  133 

where  y  is  expressed  in  C.G.S.  units.     Hence,  if  y  is  ex- 
pressed in  practical  units  or  10  ~9  C.G.S.  units, 

c  =  7rVn°'-  =  1.645  </2  10  -9. 

Substituting  for  the  conductivity  of  sheet  iron  the  ap- 

proximate value, 

y  =  105, 

we  get  as  the  coefficient  of  eddy  currents  for  laminated  iron, 
2-»=  1.645</210-9- 


loss  of  energy  per  cm3  and  cycle, 

W=  ey^Wfc2  =  -  //2y^(B210-9  =  1.645  </2y  N<$?  10  ~9  ergs 
6 

=  1.645</27V~(B210-4ergs; 
or,        W  =  c  y  NW  10  -  7  =  1.645  d*  N  <S?  10  -  "  joules  ; 

loss  of  power  per  cm3  at  frequency,  N, 

p  =  NW  '=  cy^2«210-7  =  1.645  </W2(B2  10  ~n  watts; 
total  loss  of  power  in  volume,   V, 

p  =  vp  =  1.645  ^/2^2(B210-n  watts. 

As  an  example, 

d  =  1  mm  =  .1  cm  ;  N=  100  ;  OS  =  5000;    V  =  1000  cm8. 
e  =  1,645  X  10-"; 
^F=  4110  ergs 

=  .000411  joules; 
/  =  .0411  watts; 
P  =  41.1  watts. 

90.    (6):  Iron  Wire. 

Let,  in  Fig.  80,  d  = 
diameter  of  a  piece  of 
iron  wire  ;  then  if  x  is 
the  radius  of  a  circular 
zone  of  thickness,  d  x, 
and  one  cm  in  length, 
the  conductance  of  this  pig.  so. 


134  ALTERNATING-CURRENT  PHENOMENA. 

zone  is,  y^/x/2  TT  x,  and  the  magnetic  flux  inclosed  by  the 
zone  is  (B  x2  *. 

Hence,  the  E.M.F.  induced  in  this  zone  is  : 

8£  =  V2  7r2^(B  x2,  in  C.G.S.  units, 
and  the  current  produced  thereby  is, 


,  in  C.G.S.  units. 

The  power  consumed  in  this  zone  is,  therefore, 

dP=  §EdI  =  7T8  y  N'2  (B2  x3  d  x,  in  C.G.S.  units 

consequently,  the  total  power  consumed  in  one  cm  length 
of  wire  is 

8  P  =   f~  dW  =  7T3  y  N'1  ®2  f  *  xa  dx 

=  ^-y^2&V4,  in  C.G.S.  units. 
Since  the  volume  of  one  cm  length  of  wire  is 

/  ,*?,  -     'I 

the  power  consumed  in  one  cm3  of  iron  is 

x  P        2 

P  =  -^-  =  ^  y  ^2(BV2,  in  C.G.S.  units  or  erg-seconds, 

and  the  energy  consumed  per  cycle  and  cm3  of  iron  is 

ergs. 


Therefore,  the  coefficient  of  eddy  currents  for  iron  wire  is 
c  =  ^^2  =  .617  </2; 

or,  if  y  is  expressed  in  practical  units,  or  10  ~9  C.G.S.  units, 

c  =  -^ 
10 


FOUCAULT  OR  EDDY  CURRENTS.  135 

Substituting  ^  =  ^ 

we  get  as  the  coefficient  of  eddy  currents  for  iron  wire, 

e=  —  ^210~9  =  .617  </210-9. 
16 

The    loss    of    energy  per  cm3    of    iron,   and   per   cycle 
becomes 


=  .617  d*N®?  10~4  ergs, 

loss  of  power  per  cm3,  at  frequency,  N, 

p  =  Nh  =  ey^2(B210-7  =  .617  d 2 N*<$?  10 -"  watts; 
total  loss  of  power  in  volume,   V, 

P=  Vp  =  .617  FVJV'&'IO-11  watts. 
As  an  example, 
d  =  1  mm,  =  .1  cm  ;  N=  100  ;  «2  =  5,000 ;    V=  1000  cm8. 

e  =  .617  X  10-11, 
W=  1540  ergs  =  .000154  joules, 
p  =  .0154  watts, 
P  =  15.4  watts, 

hence  very  much  less  than  in  sheet  iron  of  equal  thickness. 

91.    Comparison  of  sheet  iron  and  iron  wire. 

If 

//!  =  thickness  of  lamination  of  sheet  iron,  and 
dz  =  diameter  of  iron  wire, 

the  eddy-coefficient  of  sheet  iron  being 
T*  j  2 10-9 

*    T? 

and  the  eddy  coefficient  of  iron  wire 


136  AL  TERNA  TING-CURRENT  PHENOMENA. 

the  loss  of  power  is  equal   in   both  —  other  things   being 
equal  —  if  ex  =  e2  ;  that  is,  if, 

#  =  !</!»,  or  4  =  1.63  ^. 
o 

It  follows  that  the  diameter  of  iron  wire  can  be  1.63 
times,  or,  roughly,  1|  as  large  as  the  thickness  of  laminated 
iron,  to  give  the  same  loss  of  energy  through  eddy  currents, 
as  shown  in  Fig.  81. 


Fig.  81. 

92.    Demagnetizing,  or  screening  effect  of  eddy  currents. 

The  formulas  derived  for  the  coefficient  of  eddy  cur- 
rents in  laminated  iron  and  in  iron  wire,  hold  only  when 
the  eddy  currents  are  small  enough  to  neglect  their  mag- 
netizing force.  Otherwise  the  phenomenon  becomes  more 
complicated;  the  magnetic  flux  in  the  interior  of  the  lam- 
ina, or  the  wire,  is  not  in  phase  with  the  flux  at  the  sur- 
face, but  lags  behind  it.  The  magnetic  flux  at  the  surface 
is  due  to  the  impressed  M.M.F.,  while  the  flux  in  the  inte- 
rior is  due  to  the  resultant  of  the  impressed  M.M.F.  and  to 
the  M.M.F.  of  eddy  currents  ;  since  the  eddy  currents  lag 
90°  behind  the  flux  producing  them,  their  resultant  with 
the  impressed  M.M.F.,  and  therefore  the  magnetism  in  the 


FOUCAULT  OR   EDDY  CUKREN7*S.  137 

interior,  is  made  lagging.  Thus,  progressing  from  the  sur- 
face towards  the  interior,  the  magnetic  flux  gradually  lags 
more  and  more  in  phase,  and  at  the  same  time  decreases 
in  intensity.  While  the  complete  analytical  solution  of  this 
phenomenon  is  beyond  the  scope  of  this  book,  a  determina- 
tion of  the  magnitude  of  this  demagnetization,  or  screening 
effect,  sufficient  to  determine  whether  it  is  negligible,  or 
whether  the  subdivision  of  the  iron  has  to  be  increased 
to  make  it  negligible,  can  be  made  by  calculating  the  maxi- 
mum magnetizing  effect,  which  cannot  be  exceeded  by  the 
eddys. 

Assuming  the  magnetic  density  as  uniform  over  the 
whole  cross-section,  and  therefore  all  the  eddy  currents  in 
phase  with  each  other,  their  total  M.M.F.  represents  the 
maximum  possible  value,  since  by  the  phase  difference  and 
the  lesser  magnetic  density  in  the  center  the  resultant 
M.M.F.  is  reduced. 

In  laminated  iron  of  thickness  d,  the  current  in  a  zone 
of  thickness,  dx  at  distance  x  from  center  of  sheet,  is  : 


dl  =         -rrN&jxdx  units  (C.G.S.) 

=  V2  TT  N&jxdx  10  -  8  amperes  ; 
hence  the  total  current  in  sheet  is 

/= 


amperes. 


Hence,  the  maximum  possible  demagnetizing  ampere-turns 
acting  upon  the  center  of  the  lamina,  are 

A/9 

-  8  =  .555  N&jd*  10  -  8 


8 
=  .555  ./V(B</210~3  ampere-turns  per  cm 

Example  :     d  =  .1  cm,     N=  100,     (B  =  5,000, 
or  /  =  2.775  ampere-turns  per  cm. 


138         ALTERNATING-CURRENT  PHENOMENA. 

93.    In  iron  wire  of  diameter  d,  the  current  in  a  tubular 
zone  of  dx  thickness  and  x  radius  is 

dl=  —  TT  JV&j'x  dxlO-*  amperes; 
hence,  the  total  current  is 

I  =    f$4I~?2.  vN&j  10-«  f*  xdx 

Jo  "  Jo 

A/9 


~  *  amperes. 
16 

Hence,  the  maximum  possible  demagnetizing  ampere-turns, 
acting  upon  the  center  of  the  wire,  are 


10  - 


16 

=  .2775  N(S>  d*  10  -  8  ampere-turns  per  cm. 

For  example,  if  d=  .1  cm,  N  =  100,  «  =  5,000,  then 
/=  1,338  ampere-turns  per  cm;  that  is,  half  as  much  as  in 
a  lamina  of  the  thickness  d. 

94.  Besides  the  eddy,  or  Foucault,  currents  proper,  which 
flow  as  parasitic  circuits  in  the  interior  of  the  iron  lamina 
or  wire,  under  certain  circumstances  eddy  currents  also 
flow  in  larger  orbits  from  lamina  to  lamina  through  the 
whole  magnetic  structure.  Obviously  a  calculation  of  these 
eddy  currents  is  possible  only  in  a  particular  structure. 
They  are  mostly  surface  currents,  due  to  short  circuits 
existing  between  the  laminae  at  the  surface  of  the  magnetic 
structure. 

Furthermore,  eddy  currents  are  induced  outside  of  the 
magnetic  iron  circuit  proper,  by  the  magnetic  stray  field 
cutting  electric  conductors  in  the  neighborhood,  especially 
when  drawn  towards  them  by  iron  masses  behind,  in  elec- 
tric conductors  passing  through  the  iron  of  an  alternating 
field,  etc.  All  these  phenomena  can  be  calculated  only  in 
particular  cases,  and  are  of  less  interest,  since  they  can 
and  should  be  avoided. 


FOUCAULT  OR  EDDY  CURRENTS.  139 

Eddy  Currents  in   Conductor,  and   Unequal  Current 
Distribution. 

95.  If  the  electric  conductor  has  a  considerable  size,  the 
alternating  magnetic  field,  in  cutting  the  conductor,  may 
set  up  differences  of  potential  between  the  different  parts 
thereof,  thus  giving  rise  to  local  or  eddy  currents  in  the 
copper.  This  phenomenon  can  obviously  be  studied  only 
with  reference  to  a  particular  case,  where  the  shape  of  the 
conductor  and  the  distribution  of  the  magnetic  field  are 
known. 

Only  in  the  case  where  the  magnetic  field  is  produced 
by  the  current  flowing  in  the  conductor  can  a  general  solu- 
tion be  given.  The  alternating  current  in  the  conductor 
produces  a  magnetic  field,  not  only  outside  of  the  conductor, 
but  inside  of  it  also ;  and  the  lines  of  magnetic  force  which 
close  themselves  inside  of  the  conductor  induce  E.M.Fs. 
in  their  interior  only.  Thus  the  counter  E.M.F.  of  self- 
inductance  is  largest  at  the  axis  of  the  conductor,  and  least 
at  its  surface ;  consequently,  the  current  density  at  the 
surface  will  be  larger  than  at  the  axis,  or,  in  extreme  cases, 
the  current  may  not  penetrate  at  all  to  the  center,  or  a 
reversed  current  flow  there.  Hence  it  follows  that  only  the 
exterior  part  of  the  conductor  may  be  used  for  the  conduc- 
tion of  the  current,  thereby  causing  an  increase  of  the 
ohmic  resistance  due  to  unequal  current  distribution. 

The  general  solution  of  this  problem  for  round  conduc- 
tors leads  to  complicated  equations,  and  can  be  found  else- 
where. 

In  practice,  this  phenomenon  is  observed  only  with  very 
high  frequency  currents,  as  lightning  discharges  ;  in  power 
distribution  circuits  it  has  to  be  avoided  by  either  keeping 
the  frequency  sufficiently  low,  or  having  a  shape  of  con- 
ductor such  that  unequal  current  distribution  does  not 
take  place,  as  by  using  a  tubular  or  a  flat  conductor,  or 
several  conductors  in  parallel. 


140  ALTERNATING-CURRENT  PHENOMENA. 

96.  It  will,  therefore,  be  sufficient  to  determine  the 
largest  size  of  round  conductor,  or  the  highest  frequency, 
where  this  phenomenon  is  still  negligible. 

In  the  interior  of  the  conductor,  the  current  density 
is  not  only  less  than  at  the  surface,  but  the  current  lags 
behind  the  current  at  the  surface,  due  to  the  increased 
effect  of  self-inductance.  This  lag  of  the  current  causes  the 
magnetic  fluxes  in  the  conductor  to  be  out  of  phase  with 
each  other,  making  their  resultant  less  than  their  sum,  while 
the  lesser  current  density  in  the  center  reduces  the  total 
flux  inside  of  the  conductor.  Thus,  by  assuming,  as  a  basis 
for  calculation,  a  uniform  current  density  and  no  difference 
of  phase  between  the  currents  in  the  different  layers  of  the 
conductor,  the  unequal  distribution  is  found  larger  than  it 
is  in  reality.  Hence  this  assumption  brings  us  on  the  safe 
side,  and  at  the  same  time  simplifies  the  calculation  greatly. 

Let  Fig.  82  represent  a  cross-section  of  a  conductor  of 
radius  R,  and  a  uniform  current  density, 


where  /  =  total  current  in  conductor. 


Fig.  82. 


The  magnetic  reluctance  of  a  tubular  zone  of  unit  length 
and  thickness  dxt  of  radius  x,  is 


FOUCAULT  OR  EDDY  CURRENTS.  141 

The  current  inclosed  by  this  zone  is  Ix  =  zW,  and  there 
fore,  the  M.M.F.  acting  upon  this  zone  is 

$x  =  47r  Ix/  10  =  4  **«»/  10, 

and  the  magnetic  flux  in  this  zone  is 

d$>  =  $x  I  G(x  =  2  Trixdx  /  10. 
Hence,  the  total  magnetic  flux  inside  the  conductor  is 

,  27T    .    CR        .  TTiR*  I 


From  this  we  get,  as  the  excess  of  counter  E.M.F.  at  the 
axis  of  the  conductor  over  that  at  the  surface  — 

&E  =  V27r^0>  10  ~8  =  V27r7W10  -9,  per  unit  length, 


and  the  reactivity,  or  specific  reactance  at  the  center  of  the 
conductor,  becomes  k  =  &E  /  i  =  V2  i^NR*  10  ~9. 
Let  p  =  resistivity,  or  specific  resistance,  of  the  material  of 
the  conductor. 

We  have  then,     k/p  =  V^TrW^lO-9/?; 
and  p/  VFT7, 

the  ratio  of  current  densities  at  center  and  at  periphery. 

For  example,  if,  in  copper,  p  =  1.7xlO—  6,  and  the 
percentage  decrease  of  current  density  at  center  shall  not 
exceed  5  per  cent,  that  is  — 

P  -H  VF+72  =  .95  -  1, 

we  have,  £  =  .51xlO-«; 

hence        .51  x  10-6=  V^TrW^lO-9 
or  N2?  =  36.6  ; 

hence,  when  N=       125        100        60     25 

£  =      .541       .605       .781     1.21  cm. 
D  =  1R=    1.08       1.21       1.56       2.42cm. 
Hence,  even  at  a  frequency  of  125  cycles,  the  effect  of 
unequal  current  distribution   is   still  negligible   at   one   cm 
diameter  of   the  conductor.     Conductors  of   this   size  are, 
however,  excluded  from  use  at  this  frequency  by  the  exter- 
nal self-induction,  which   is   several  times   larger  than  the. 


142  ALTERNATING-CURRENT  PHENOMENA. 

resistance.  We  thus  see  that  unequal  current  distribution 
is  usually  negligible  in  practice.  The  above  calculation  was 
made  under  the  assumption  that  the  conductor  consists  of 
unmagnetic  material.  If  this  is  not  the  case,  but  the  con- 
ductor of  iron  of  permeability  p.,  then  ;  d$  =  pffx  /  (&x  and 
thus  ultimately ;  k  =  V2  wW/^10  ~"  and  ;  k  /  P  =  V2  ** 
NpR*  10— '//»•  Thus,  for  instance,  for  iron  wire  at 
/>  =  10xlO-6,  ft  =  500  it  is,  permitting  5%  difference 
between  center  and  outside  of  wire;  k  =  3.2  X  10 ~6  and 
NR*  =  .46, 
hence  when,  N  =  125  100  60  25 

X  =  .061     .068     .088     .136  cm. 
thus  the  effect  is  noticeable  even  with  relatively  small  iron 

wire. 

Mutual  Inductance. 

97.  When  an  alternating  magnetic  field  of  force  includes 
a  secondary  electric  conductor,  it  induces  therein  an  E.M.F. 
which  produces  a  current,  and  thereby  consumes  energy  if 
the  circuit  of  the  secondary  conductor  is  closed. 

A  particular  case  of  such  induced  secondary  currents 
are  the  eddy  or  Foucault  currents  previously  discussed. 

Another  important  case  is  the  induction  of  secondary 
E.M.Fs.  in  neighboring  circuits ;  that  is,  the  interference  of 
circuits  running  parallel  with  each  other. 

In  general,  it  is  preferable  to  consider  this  phenomenon 
of  mutual  inductance  as  not  merely  producing  an  energy 
component  and  a  wattless  component  of  E.M.F.  in  the 
primary  conductor,  but  to  consider  explicitly  both  the  sec- 
ondary and  the  primary  circuit,  as  will  be  done  in  the 
chapter  on  the  alternating-current  transformer. 

Only  in  cases  where  the  energy  transferred  into  the 
secondary  circuit  constitutes  a  small  part  of  the  total  pri- 
mary energy,  as  in  the  discussion  of  the  disturbance  caused 
by  one  circuit  upon  a  parallel  circuit,  may  the  effect  on  the 
primary  circuit  be  considered  analogously  as  in  the  chapter 
•on  eddy  currents,  by  the  introduction  of  an  energy  com- 


FOUCAULT  OR  EDDY  CURRENTS.  143 

ponent,  representing  the  loss  of  power,  and  a  wattless 
component,  representing  the  decrease  of  self-inductance. 

Let  — 

x  =  2  TT  N L  =  reactance  of  main  circuit ;  that  is,  L  = 
total  number  of  interlinkages  with  the  main  conductor,  of 
the  lines  of  magnetic  force  produced  by  unit  current  in 
that  conductor  ; 

.#!  =  2-jrNL1  =  reactance  of  secondary  circuit ;  that  is, 
Ll  =  total  number  of  interlinkages  with  the  secondary 
conductor,  of  the  lines  of  magnetic  force  produced  by  unit 
current  in  that  conductor ; 

xm  =  2  TT  N  Lm  =  mutual  inductance  of  circuits  ;  that  is, 
Lm  =  total  number  of  interlinkages  with  the  secondary 
conductor,  of  the  lines  of  magnetic  force  produced  by  unit 
current  in  the  main  conductor,  or  total  number  of  inter- 
linkages with  the  main  conductor  of  the  lines  of  magnetic 
force  produced  by  unit  current  in  the  secondary  conductor. 
Obviously  :  xm*  <  xx^* 

*  As  coefficient  of  self-inductance  L,  L^,  the  total  flux  surrounding  the  conductor 
is  here  meant.  Usually  in  the  discussion  of  inductive  apparatus,  especially  of  trans- 
formers, that  part  of  the  magnetic  flux  is  derroted  self-inductance  of  the  one  circuit 
which  surrounds  this  circuit,  but  not  the  other  circuit ;  that  is,  which  passes  between 
both  circuits.  Hence,  the  total  self-inductance,  L,  is  in  this  ease  equal  to  the  sum  of 
the  self-inductance,  Z,j,  and  the  mutual  inductance,  Lm. 

The  object  of  this  distinction  is  to  separate  the  wattless  part,  Z1?  of  the 
total  self-inductance,  L,  from  that  part,  Lm,  which  represents  the  transfer  of 
E.M.F.  into  the  secondary  circuit,  since  the  action  of  these  two  components  is 
essentially  different. 

Thus,  in  alternating-current  transformers  it  is  customary  —  and  will  be 
done  later  in  this  book  —  to  denote  as  the  self-inductance,  Z,  of  each  circuit 
only  that  part  of  the  magnetic  flux  produced  by  the  circuit  which  passes 
between  both  circuits,  and  thus  acts  in  "  choking  "  only,  but  not  in  transform- 
ing; while  the  flux  surrounding  both  circuits  is  called  mutual  inductance,  or 
useful  magnetic  flux. 

With'  this  denotation,  in  transformers  the  mutual  inductance,  Lm,  is  usu- 
ally very  much  greater  than  the  self-inductances,  //,  and  Z/,  while,  if  the 
self-inductances,  Z  and  Zj ,  represent  the  total  flux,  their  product  is  larger 
than  the  square  of  the  mutual  inductance,  Lm  ;  or 


144  ALTERNATING— CURRENT  PHENOMENA. 

Let  rx  =  resistance  of  secondary  circuit.     Then  the  im- 
pedance of  secondary  circuit  is 

^i  =  rv  —  /*! ,  zl  =  V/v  +  xi2 ; 

E.M.F.  induced  in  the  secondary  circuit,  £±  =  jxmf, 
where  /  =  primary  current.   Hence,  the  secondary  current  is 


and  the  E.M.F.  induced  in  the  primary  circuit  by  the  secon- 
dary current,  7l  is 


or,  expanded, 

Y   zr  j~.   2 

xm^          JXm 


2  _i_  r  2        r2    i    JT 

T^  ^i  "l       "    •*  2 

Hence,  the  E.M.F.  consumed  thereby 


effective  resistance  of  mutual  inductance ; 


^  =  effective  reactance  of  mutual  inductance. 

The  susceptance  of  mutual  inductance  is  negative,  or  of 
opposite  sign  from  the  reactance  of  self-inductance.  Or, 

Mutual  inductance  consumes  energy  and  decreases  the  self- 
inductance. 

Dielectric  and  Electrostatic  Phenomena. 
98.  While  magnetic  hysteresis  and  eddy  currents  can 
be  considered  as  the  energy  component  of  inductance,  con- 
densance  has  an  energy  component  also,  namely,  dielectric 
hysteresis.  In  an  alternating  magnetic  field,  energy  is  con- 
sumed in  hysteresis  due  to  molecular  friction,  and  similarly, 
energy  is  also  consumed  in  an  alternating  electrostatic  field 
in  the  dielectric  medium,  in  what  is  called  electrostatic  or 
dielectric  hysteresis. 


FOUCAULT  OR   EDDY  CURRENTS.  145 

While  the  laws  of  the  loss  of  energy  by  magnetic  hys- 
teresis are  fairly  well  understood,  and  the  magnitude  of  the 
effect  known,  the  phenomenon  of  dielectric  hysteresis  is 
still  almost  entirely  unknown  as  concerns  its  laws  and  the 
magnitude  of  the  effect. 

It  is  quite  probable  that  the  loss  of  power  in  the  dielec- 
tric in  an  alternating  electrostatic  field  consists  of  two  dis- 
tinctly different  components,  of  which  the  one  is  directly 
proportional  to  the  frequency,  —  analogous  to  magnetic 
hysteresis,  and  thus  a  constant  loss  of  energy  per  cycle, 
independent  of  the  frequency ;  while  the  other  component 
is  proportional  to  the  square  of  the  frequency,  —  analogous 
to  the  loss  of  power  by  eddy  currents  in  the  iron,  and  thus 
a  loss  of  energy  per  cycle  proportional  to  the  frequency. 

The  existence  of  a  loss  of  power  in  the  dielectric,  pro- 
portional to  the  square  of  the  frequency,  I  observed  some 
time  ago  in  paraffined  paper  in  a  high  electrostatic  field  and 
at  high  frequency,  by  the  electro-dynamometer  method, 
and  other  observers  under  similar  conditions  have  found 
the  same  result. 

Arno  of  Turin  found  at  low  frequencies  and  low  field 
strength  in  a  larger  number  of  dielectrics,  a  loss  of  energy 
per  cycle  independent  of  the  frequency,  but  proportional  to 
the  1.6th  power  of  the  field  strength,  —  that  is,  following 
the  same  law  as  the  magnetic  hysteresis, 

^  =  ^(B'-6. 

This  loss,  probably  true  dielectric  static  hysteresis,  was 
observed  under  conditions  such  that  a  loss  proportional  to 
the  square  of  density  and  frequency  must  be  small,  while  at 
high  densities  and  frequencies,  as  in  condensers,  the  true 
dielectric  hysteresis  may  be  entirely  obscured  by  a  viscous 
loss,  represented  by  W^  =  e7V(B2. 

99.  If  the  loss  of  power  by  electrostatic  hysteresis  is 
proportional  to  the  square  of  the  frequency  and  of  the  field 
intensity,  —  as  it  probably  nearly  is  under  the  working  con- 


146  AL  TERNA  TING-CURRENT  PHENOMENA. 

ditions  of  alternating-current  condensers,  —  then  it  is  pro- 
portional to  the  square  of  the  E.M.F.,  that  is,  the  effective 
conductance,  g,  due  to  dielectric  hysteresis  is  a  constant ; 
and,  since  the  condenser  susceptance,  —  b=  b',  is  a  constant 
also,  —  unlike  the  magnetic  inductance,  —  the  ratio  of  con- 
ductance and  susceptance,  that  is,  the  angle  of  difference 
of  phase  due  to  dielectric  hysteresis,  is  a  constant.  This  I 
found  proved  by  experiment.  This  would  mean  that  the 
dielectric  hysteretic  admittance  of  a  condenser, 

Y=g+jb=g-jb', 

where  :  g  =  hysteretic  conductance,  b'  =  hysteretic  suscep- 
tance ;  and  the  dielectric  hysteretic  impedance  of  a  con- 
denser, „  .  .  . 

Z  =  r  —  jx  —  r  +jxc, 

where  :  r  =  hysteretic  resistance,  xc  —  hysteretic  condens- 
ance  ;  and  the  angle  of  dielectric  hysteretic  lag,  tan  a  =  b'  / g 
=  xc  /  r,  are  constants  of  the  circuit,  independent  of  E.M.F. 
and  frequency.  The  E.M.F.  is  obviously  inversely  propor- 
tional to  the  frequency. 

The  true  static  dielectric  hysteresis,  observed  by  Arno 
as  proportional  to  the  1.6th  power  of  the  density,  will  enter 
the  admittance  and  the  impedance  as  a  term  variable  and 
dependent  upon  E.M.F.  and  frequency,  in  the  same  manner 
as  discussed  in  the  chapter  on  magnetic  hysteresis. 

To  the  magnetic  hysteresis  corresponds,  in  the  electro- 
static field,  the  static  component  of  dielectric  hysteresis, 
following,  probably,  the  same  law  of  1.6th  power. 

To  the  eddy  currents  in  the  iron  corresponds,  in  the 
electrostatic  field,  the  viscous  component  of  dielectric  hys- 
teresis, following  the  square  law. 

As  a  rule  however,  these  hysteresis  losses  in  the  alter- 
nating electrostatic  field  of  a  condenser  are  very  much 
smaller  than  the  losses  in  an  alternating  magnetic  field,  so 
that  while  the  latter  exert  a  very  marked  effect  on  the  de- 
sign of  apparatus,  representing  frequently  the  largest  of  all 
the  losses  of  energy,  the  dielectric  losses  are  so  small  as  to 
be  very  difficult  to  observe. 


FOUCAULT  OR   EDDY  CURRENTS.  147 

To  the  phenomenon  of  mutual  inductance  corresponds, 
in  the  electrostatic  field,  the  electrostatic  induction,  or  in- 
fluence. 

100.  The  alternating  electrostatic  field  of  force  of  an 
electric  circuit  induces,  in  conductors  within  the  field  of 
force,  electrostatic  charges  by  what  is  called  electrostatic 
influence.  These  charges  are  proportional  to  the  field 
strength  ;  that  is,  to  the  E.M.F.  in  the  main  circuit. 

If  a  flow  of  current  is  produced  by  the  induced  charges, 
energy  is  consumed  proportional  to  the  square  of  the  charge  ; 
that  is,  to  the  square  of  the  E.M.F. 

These  induced  charges,  reacting  upon  the  main  conduc- 
tor, influence  therein  charges  of  equal  but  opposite  phase, 
and  hence  lagging  behind  the  main  E.M.F.  by  the  angle 
of  lag  between  induced  charge  and  inducing  field.  They 
require  the  expenditure  of  a  charging  current  in  the  main 
conductor  in  quadrature  with  the  induced  charge  thereon ; 
that  is,  nearly  in  quadrature  with  the  E.M.F.,  and  hence 
consisting  of  an  energy  component  in  phase  with  the 
E.M.F.  —  representing  the  power  consumed  by  electrostatic 
influence  —  and  a  wattless  component,  which  increases  the 
capacity  of  the  conductor,  or,  in  other  words,  reduces  its 
capacity  reactance,  or  condensance. 

Thus,  the  electrostatic  influence  introduces  an  effective 
conductance,  g,  and  an  effective  susceptance,  b,  —  of  the 
same  sign  with  condenser  susceptance,  —  into  the  equations 
of  the  electric  circuit. 

While  theoretically  g  and  b  should  be  constants  of  the 
circuit,  frequently  they  are  very  far  from  such,  due  to 
disruptive  phenomena  beginning  to  appear  at  high  electro- 
static stresses. 

Even  the  capacity  condensance  changes  at  very  high 
potentials  ;  escape  of  electricity  into  the  air  and  over  the 
surfaces  of  the  supporting  insulators  by  brush  discharge  or 
electrostatic  glow  takes  place.  As  far  as  this  electrostatic 


148  ALTERNATING-CURRENT  PHENOMENA 

corona  reaches,  the  space  is  in  electric  connection  with  the 
conductor,  and  thus  the  capacity  of  the  circuit  is  deter- 
mined, not  by  the  surface  of  the  metallic  conductor,  but 
by  the  exterior  surface  of  the  electrostatic  glow  surround- 
ing the  conductor.  This  means  that  with  increasing  po- 
tential, the  capacity  increases  as  soon  as  the  electrostatic 
corona  appears  ;  hence,  the  condensance  decreases,  and  at 
the  same  time  an  energy  component  appears,  representing 
the  loss  of  power  in  the  corona. 

This  phenomenon  thus  shows  some  analogy  with  the  de- 
crease of  magnetic  inductance  due  to  saturation. 

At  moderate  potentials,  the  condensance  due  to  capacity 
can  be  considered  as  a  constant,  consisting  of  a  wattless 
component,  the  condensance  proper,  and  an  energy  com- 
ponent, the  dielectric  hysteresis. 

The  condensance  of  a  polarization  cell,  however,  begins 
to  decrease  at  very  low  potentials,  as  soon  as  the  counter 
E.M.F.  of  chemical  dissociation  is  approached. 

The  condensance  of  a  synchronizing  alternator  is  of 
the  nature  of  a  variable  quantity ;  that  is,  the  effective 
reactance  changes  gradually,  according  to  the  relation  of 
impressed  and  of  counter  E.M.F.,  from  inductance  over 
zero  to  condensance. 

Besides  the  phenomena  discussed  in  the  foregoing  as 
terms  of  the  energy  components  and  the  wattless  compo- 
nents of  current  and  of  E.M.F.,  the  electric  leakage  is 
to  be  considered  as  a  further  energy  component ;  that  is, 
the  direct  escape  of  current  from  conductor  to  return  con- 
ductor through  the  surrounding  medium,  due  to  imperfect 
insulating  qualities.  This  leakage  current  represents  an 
effective  conductance,  g,  theoretically  independent  of  the 
E.M.F.,  but  in  reality  frequently  increasing  greatly  with  the 
E.M.F.,  owing  to  the  decrease  of  the  insulating  strength  of 
the  medium  upon  approaching  the  limits  of  its  disruptive 
strength. 


•     FOUCAULT  OR  EDDY  CURRENTS.  149 

101.  In  the  foregoing,  the  phenomena  causing  loss  of 
energy  in  an  alternating-current  circuit  have  been  dis- 
cussed ;  and  it  has  been  shown  that  the  mutual  relation 
between  current  and  E.M.F.  can  be  expressed  by  two  of 
the  four  constants  : 

Energy   component  of   E.M.F.,  in  phase  with  current,  and  = 

current  X  effective  resistance,  or  r ; 
wattless  component  of  E.M.F.,  in  quadrature  with  current,  and  = 

current 'X  effective  reactance,  or  x  • 
energy   component   of   current,  in  phase  with   E.M.F.,  and  = 

E.M.F.  X  effective  conductance,  or  g ; 
wattless  component  of  current,  in  quadrature  with  E.M.F.,  and  = 

E.M.F.  X  effective  susceptance,  or  b. 

In  many  cases  the  exact  calculation  of  the  quantities, 
r,  x,  g,  b,  is  not  possible  in  the  present  state  of  the  art. 

In  general,  r,  x,  g,  b,  are  not  constants  of  the  circuit,  but 
depend  —  besides  upon  the  frequency  —  more  or  less  upon 
E.M.F.,  current,  etc.  Thus,  in  each  particular  case  it  be- 
comes necessary  to  discuss  the  variation  of  r,  x,  g,  b,  or  to 
determine  whether,  and  through  what  range,  they  can  be 
assumed  as  constant. 

In  what  follows,  the  quantities  r,  x,  g,  b,  will  always  be 
considered  as  the  coefficients  of  the  energy  and  wattless 
components  of  current  and  E.M.F.,  —  that  is,  as  the  effec- 
tive quantities,  —  so  that  the  results  are  directly  applicable 
to  the  general  electric  circuit  containing  iron  and  dielectric 
losses. 

Introducing  now,  in  Chapters  VII.  to  IX.,  instead  of 
"  ohmic  resistance,"  the  term  "  effective  resistance,"  etc., 
as  discussed  in  the  preceding  chapter,  the  results  apply 
also  —  within  the  range  discussed  in  the  preceding  chapter 
—  to  circuits  containing  iron  and  other  materials  producing 
energy  losses  outside  of  the  electric  conductor. 


150  ALTERNATING-CURRENT  PHENOMENA. 


CHAPTER   XII. 

POWER,  AND   DOUBLE   FREQUENCY   QUANTITIES 
IN  GENERAL. 

102.  Graphically  alternating  currents  and  E.M.F's 
are  represented  by  vectors,  of  which  the  length  represents 
the  intensity,  the  direction  the  phase  of  the  alternating 
wave.  The  vectors  generally  issue  from  the  center  of 
co-ordinates. 

In  the  topographical  method,  however,  which  is  more 
convenient  for  complex  networks,  as  interlinked  polyphase 
circuits,  the  alternating  wave  is  represented  by  the  straight 
line  between  two  points,  these  points  representing  the  abso- 
lute values  of  potential  (with  regard  to  any  reference  point 
chosen  as  co-ordinate  center)  and  their  connection  the  dif- 
ference of  potential  in  phase  and  intensity. 

Algebraically  these  vectors  are  represented  by  complex 
quantities.  The  impedance,  admittance,  etc.,  of  the  circuit 
is  a  complex  quantity  also,  in  symbolic  denotation. 

Thus  current,  E.M.F.,  impedance,  and  admittance  are 
related  by  multiplication  and  division  of  complex  quantities 
similar  as  current,  E.M.F.,  resistance,  and  conductance  are 
related  by  Ohms  law  in  direct  current  circuits. 

In  direct  current  circuits,  power  is  the  product  of  cur- 
rent into  E.M.F.  In  alternating  current  circuits,  if 


The  product, 

P0  =  EI=  (Ml  -  *"/")  +j  (W 


POWER,   AND  DOUBLE  FREQUENCY  QUANTITIES.     151 

is  not  the  power;  that  is,  multiplication  and  division,  which 
are  correct  in  the  inter-relation  of  current,  E.M.F.,  impe- 
dance, do  not  give  a  correct  result  in  the  inter-relation  of 
E.M.F.,  current,  power.  The  reason  is,  that  El  are  vec- 
tors of  the  same  frequency,  and  Z  a  constant  numerical 
factor  which  thus  does  not  change  the  frequency. 

The  power  P,  however,  is  of  double  frequency  compared 
with  E  and  /,  that  is,  makes  a  complete  wave  for  every 
half  wave  of  E  or  7,  and  thus  cannot  be  represented  by  a 
vector  in  the  same  diagram  with  E  and  /. 

P0  =  E  I  is  a  quantity  of  the  same  frequency  with  E 
and  /,  and  thus  cannot  represent  the  power. 

\ 

103.  Since  the  power  is  a  quantity  of  double  frequency 
of  E  and  /,  and  thus  a  phase  angle  w  in  E  and  /  corre- 
sponds to  a  phase  angle  2  w  in  the  power,  it  is  of  interest  to 
investigate  the  product  E  I  formed  by  doubling  the  phase 
angle. 

Algebraically  it  is, 

P=EI=  (*  +>")  (V1  +/z  n)  = 


Since  j*  =  -  1,  that  is  180°  rotation  for  E  and  /,  for  the 
double  frequency  vector,  P,j*  =  +  1,  or  360°  rotation,  and 

j  x  1  =j 
1  x>=  -j 

That  is,  multiplication  with  /  reverses  the  sign,  since  it 
denotes  a  rotation  by  180°  for  the  power,  corresponding  to 
a  rotation  of  90°  for  E  and  /. 

Hence,  substituting  these  values,  we  have, 

p  =  [El]  =  (W1  +  ^V11)  +/  (W1  -  A'u) 

The  symbol  [E  /]  here  denotes  the  transfer  from  the 
frequency  of  E  and  /  to  the  double  frequency  of  P. 


152  AL  TERNA  TING-CURRENT  PHENOMEMA. 

The   product,  P  =  \E  /]  consists  of  two  components ; 
the  real  component, 

JP1  =  [EIJ  =  (W1  +  e"in) 
and  the  imaginary  component, 

JPJ  =j 
The  component, 

P1 

is  the  power  of  the  circuit,  =  E  I  cos  (E  /) 
The  component, 
PJ  = 


is  what  may  be  called  the  "  wattless  power,"  or  the  power- 
less or  quadrature  volt-amperes  of  the  circuit,  =  E  /sin 
(El}. 

The  real  component  will  be  distinguished  by  the  index 
1,  the  imaginary  or  wattless  component  by  the  index/. 

By  introducing  this  symbolism,  the  power  of  an  alternat- 
ing circuit  can  be  represented  in  the  same  way  as  in  the 
direct  current  circuit,  as  the  symbolic  product  of  current 
and  E.M.F. 

Just  as  the  symbolic  expression  of  current  and  E.M.F. 
as  complex  quantity  does  not  only  give  the  mere  intensity, 
but  also  the  phase, 

£  = 

jfc    == 

P 
tan  <f>  =  -j 

so  the  double  frequency  vector  product  P  =  [E  /]  denotes 
more  than  the  mere  power,  by  giving  with  its  two  compo- 
nents P1  =  [E  I]1  and  PJ  =  [E  /]•>,  the  true  energy  volt- 
amperes,  and  the  wattless  volt-amperes. 

If 

E  = 


POWER,   AND  DOUBLE  FREQUENCY  QUANTITIES.     153 

then 


and 

P1  = 


or 


2  2  22  22  22  22 

+PJ     =<*  ,1     +  *"  / 


where  ^  =  total  volt  amperes  of  circuit.     That  is, 

The  true  power  P1  and  the  wattless  power  P$  are  the  two 
rectangular  components  of  the  total  apparent  power  Q  of  the 
circuit. 

Consequently, 

In  symbolic  representation  as  double  freqi'ency  vector  pro- 
ducts, powers  can  be  combined  and  resolved  by  the  parallelo- 
gram of  vectors  just  as  currents  and  E.M.F's  in  graphical 
or  symbolic  representation. 

The  graphical  methods  of  treatment  of  alternating  cur- 
rent phenomena  are  here  extended  to  include  double  fre- 
quency quantities  as  power,  torque,  etc. 

P1 

—  =p  =  cos  w  =  power  factor. 

PJ 

—  =  q  =  sin  w  =  inductance  factor 

of  the  circuit,  and  the  general  expression  of  power  is, 


=  Q  (cos  co  -\-j  sin  o>) 

104.     The  introduction  of  the  double  frequency  vector 
product  P  =  \E  I~\  brings  us  outside  of  the  limits  of  alge- 


154  ALTERNATING-CURRENT  PHENOMENA. 

bra,  however,    and  the  commutative  principle  of  algebra, 
a  X  b  =  b  X  a,  does  not  apply  any  more,  but  we  have, 

[El]  unlike  [IE] 
since 


we  have 

[EIJ  =  [IEJ 

[EI]J=-[IE]J 

that  is,  the  imaginary  component  reverses  its  sign  by  the 
interchange  of  factors. 

The  physical  meaning  is,  that  if  the  wattless  power 
[E  7p  is  lagging  with  regard  to  E,  it  is  leading  with  regard 
to/. 

The  wattless  component  of  power  is  absent,  or  the  total 
apparent  power  is  true  power,  if 


[EI]J  =  (W1  -  A'11)  =  0. 
that  is, 


or, 

tan  (E)  =  tan  (/), 

that  is,  E  and  /  are  in  phase  or  in  opposition. 

The  true  power  is  absent,  or  the  total  apparent  power 
wattless,  if 

[El]1  =  (W1  +  M*  =  0 

that  is, 

*"  _       i1 

7  ~  ~/» 
or, 

tan  E  =  —  cot  I 

that  is,  E  and  /  are  in  quadrature, 


POWER,   AND  DOUBLE  FREQUENCY  QUANTITIES.     155 

The  wattless  power  is  lagging  (with  regard  to  E  or  lead- 
ing with  regard  to  /)  if, 


and  leading  if, 

The  true  power  is  negative,  that  is,  power  returns,  if, 


We  have, 

[£,  -  7]  =  [-  E,  7]  =  - 


that  is,  when  representing  the  power  of  a  circuit  or  a  part  of 
a  circuit,  current  and  E.M.F.  must  be  considered  in  their 
proper  relative  phases,  but  their  phase  relation  with  the  re- 
maining part  of  the  circuit  is  immaterial. 
We  have  further 

\EJT\  =  -j  [£,  7]  =  [E,  iy  -j  \E,  7]1 
\JE,  7]  =j  [E,  7]  =  -  [E,  Jy  +j  [E,  7]1 
\jEjr\  =  [£,  7]  =  [E7?  +j  [E,  jy 


105.     If      7-  =  [^/J,        7>2  =  [E2/2]  .  .  .  Pn  =  [Enln} 

are  the  symbolic  expressions  of  the  power  of  the  different 
parts  of  a  circuit  or  network  of  circuits,  the  total  power  of 
the  whole  circuit  or  network  of  circuits  is 


7^'  =  TV  +  T'ijJ.  .  •  •  +  TV 

In  other  words,  the  total  power  in  symbolic  expression 
(true  as  well  as  wattless)  of  a  circuit  or  system  is  the  sum 
of  the  powers  of  its  individual  components  in  symbolic 
expression. 

The  first  equation  is  obviously  directly  a  result  from  the 
law  of  conservation  of  energy. 


156  ALTERNATING-CURRENT  PHENOMENA. 

One  result  derived  herefrom  is  for  instance : 
If  in  a  generator  supplying  power  to  a  system  the  cur- 
rent is  out  of  phase  with  the  E.M.F.  so  as  to  give  the  watt- 
less power  Pi,  the  current  can  be  brought  into  phase  with 
the  generator  E.M.F.,  or  the  load  on  the  generator  made 
non-inductive  by  inserting  anywhere  in  the  circuit  an  appa- 
ratus producing  the  wattless  power  —  F$\  that  is,  compen- 
sation for  wattless  currents  in  a  system  takes  place  regardless 
of  the  location  of  the  compensating  device. 

Obviously  between  the  compensating  device  and  the 
source  of  wattless  currents  to  be  compensated  for,  wattless 
currents  will  flow,  and  for  this  reason  it  may  be  advisable 
to  bring  the  compensator  as  near  as  possible  to  the  circuit 
to  be  compensated. 

106.  Like  power,  torque  in  alternating  apparatus  is  a 
double  frequency  vector  product  also,  of  magnetism  and 
M.M.F.  or  current,  and  thus  can  be  treated  in  the  same 
way. 

In  an  induction  motor,  for  instance,  the  torque  is  the 
product  of  the  magnetic  flux  in  one  direction  into  the  com- 
ponent of  secondary  induced  current  in  phase  with  the 
magnetic  flux  in  time,  but  in  quadrature  position  therewith 
in  space,  times  the  number  of  turns  of  this  current,  or  since 
the  induced  E.M.F.  is  in  quadrature  and  proportional  to 
the  magnetic  flux  and  the  number  of  turns,  the  torque 
of  the  induction  motor  is  the  product  of  the  induced  E.M.F. 
into  the  component  of  secondary  current  in  quadrature 
therewith  in  time  and  space,  or  the  product  of  the  induced 
current  into  the  component  of  induced  E.M.F.  in  quadra- 
ture therewith  in  time  and  space. 

Thus  if 

E1  =  £  +jea-  —  induced  E.M.F.  in  one  direction  in 
space. 

72  =  z1  +j  z11  =  secondary  current  in  the  quadrature  di- 
rection in  space, 


POWER,   AND  DOUBLE  FREQUENCY  QUANTITIES.     157 

the  torque  is 


By  this  equation  the  torque  is  given  in  watts,  the  mean- 
ing being  that  T  =  \E  /]•>'  is  the  power  which  would  be 
exerted  by  the  torque  at  synchronous  speed,  or  the  torque 
in  synchronous  watts. 

The  torque  proper  is  then 


where 

/  =  number  of  pairs  of  poles  of  the  motor. 

In  the  polyphase  induction  motor,  if  72  =  il  +/zu  is 
the  secondary  current  in  quadrature  position,  in  space,  to 
E.M.F.  Ej. 

The  current  in  the  same  direction  in  space  as  El  is 
/!  =y72  =  —  z11  +//1;  thus  the  torque  can  also  be    ex- 
pressed as 


158  ALTERNATING-CURRENT  PHENOMENA. 


CHAPTER   XIII. 

DISTRIBUTED  CAPACITY,   INDUCTANCE,   RESISTANCE,   AND 
LEAKAGE. 

107.  As  far  as  capacity  has  been  considered  in  the 
foregoing  chapters,  the  assumption  has  been  made  that  the 
condenser  or  other  source  of  negative  reactance  is  shunted 
across  the  circuit  at  a  definite  point.  In  many  cases,  how- 
ever, the  capacity  is  distributed  over  the  whole  length  of  the 
conductor,  so  that  the  circuit  can  be  considered  as  shunted 
by  an  infinite  number  of  infinitely  small  condensers  infi 
nitely  near  together,  as  diagrammatically  shown  in  Fig.  83. 


iiiimiiiiumiiiT 

TTTTTTTTTT.TTTTTTTTTT 


i 


Fig.  83.     Distributed  Capacity. 

In  this  case  the  intensity  as  well  as  phase  of  the  current, 
and  consequently  of  the  counter  E.M.F.  of  inductance  and 
resistance,  vary  from  point  to  point ;  and  it  is  no  longer 
possible  to  treat  the  circuit  in  the  usual  manner  by  the 
vector  diagram. 

This  phenomenon  is  especially  noticeable  in  long-distance 
lines,  in  underground  cables,  and  to  a  certain  degree  in  the 
high-potential  coils  of  alternating-current  transformers  for 
very  high  voltage.  It  has  the  effect  that  not  only  the 
E.M.Fs.,  but  also  the  currents,  at  the  beginning,  end,  and 
different  points  of  the  conductor,  are  different  in  intensity 
and  in  phase. 

Where  the  capacity  effect  of  the  line  is  small,  it  may 
with  sufficient  approximation  be  represented  by  one  con- 


DISTRIBUTED    CAPACITY.  159 

denser  of  the  same  capacity  as  the  line,  shunted  across  the 
line.  Frequently  it  makes  no  difference  either,  whether 
this  condenser  is  considered  as  connected  across  the  line  at 
the  generator  end,  or  at  the  receiver  end,  or  at  the  middle. 

The  best  approximation  is  to  consider  the  line  as  shunted 
at  the  generator  and  at  the  motor  end,  by  two  condensers 
of  \  the  line  capacity  each,  and  in  the  middle  by  a  con- 
denser of  |  the  line  capacity.  This  approximation,  based 
on  Simpson's  rule,  assumes  the  variation  of  the  electric 
quantities  in  the  line  as  parabolic.  If,  however,  the  capacity 
of  the  line  is  considerable,  and  the  condenser  current  is  of 
the  same  magnitude  as  the  main  current,  such  an  approxi- 
mation is  not  permissible,  but  each  line  element  has  to  be 
considered  as  an  infinitely  small  condenser,  and  the  differ- 
ential equations  based  thereon  integrated.  Or  the  pheno- 
mena occurring  in  the  circuit  can  be  investigated  graphically 
by  the  method  given  in  Chapter  VI.  §  37,  by  dividing  the 
circuit  into  a  sufficiently  large  number  of  sections  or  line 
elements,  and  then  passing  from  line  element  to  line  element, 
to  construct  the  topographic  circuit  characteristics. 

108.  It  is  thus  desirable  to  first  investigate  the  limits 
of  applicability  of  the  approximate  representation  of  the  line 
by  one  or  by  three  condensers. 

Assuming,  for  instance,  that  the  line  conductors  are  of 
1  cm.  diameter,  and  at  a  distance  from  each  other  of  50  cm., 
and  that  the  length  of  transmission  is  50  km.,  we  get  the 
capacity  of  the  transmission  line  from  the  formula  — 

C  =  1.11  X  10  -«K/  -=-  4  loge  2  d/  8  microfarads, 
where 

K  =  dielectric  constant  of  the  surrounding  medium  =  1  in  air ; 

/  =  length  of  conductor  =  5  x  106  cm. ; 

d  =  distance  of  conductors  from  each  other  =  50  cm. ; 

8  =  diameter  of  conductor  =  1  cm. 

Since  C  =  .3  microfarads, 

the  capacity  reactance  is  x  —  106  /  2  TT  NC  ohms, 


160  ALTERNATING-CURRENT  PHENOMENA. 

where  N '=  frequency;  hence,  at  N  =  60  cycles, 

x  =  8,900  ohms  ; 

and  the  charging  current  of  the  line,  at  E  =  20,000  volts, 
becomes,  ^  =  E  /  x  =  2.25  amperes. 

The  resistance  of  100  km  of  line  of  1  cm  diameter  is  22 
ohms  ;  therefore,  at  10  per  cent  =  2,000  volts  loss  in  the 
line,  the  main  current  transmitted  over  the  line  is 

2,000 
/  =  -^-  =  91  amperes, 

representing  about  1,800  kw. 

In  this  case,  the  condenser  current  thus  amounts  to  less 
than  2^  per  cent.,  and  hence  can  still  be  represented  by  the 
approximation  of  one  condenser  shunted  across  the  line. 

If  the  length  of  transmission  is  150  km.,  and  the  voltage, 
30,000, 

capacity  reactance  at  60  cycles,  x  =  2,970  ohms ; 

charging  current,  i0  =  10.1  amperes  ; 

line  resistance,  r  =  66  ohms ; 

main  current  at  10  per  cent  loss,  7=  45.5  amperes. 

The  condenser  current  is  thus  about  22  per  cent  of  the 
main  current,  and  the  approximate  calculation  of  the  effect 
of  line  capacity  still  fairly  accurate. 

At  300  km  length  of  transmission  it  will,  at  10  per  cent, 
loss  and  with  the  same  size  of  conductor,  rise  to  nearly  90 
per  cent,  of  the  main  current,  thus  making  a  more  explicit 
investigation  of  the  phenomena  in  the  line  necessary. 

In  most  cases  of  practical  engineering,  however,  the  ca- 
pacity effect  is  small  enough  to  be  represented  by  the  approx- 
imation of  one  ;  viz.,  three  condensers  shunted  across  the  line. 

109.  A.}  Line  capacity  represented  by  one  condenser 
shunted  across  middle  of  line. 

Let  — 

Y  =  g  +  j b  =  admittance  of  receiving  circuit ; 
z  =  r  —  j  x  =  impedance  of  line ; 
be  =  condenser  susceptance  of  line. 


DISTRIBUTED   CAPACITY. 


161 


Denoting,  in  Fig.  84, 

the  E.M.F.,  viz.,  current  in  receiving  circuit  by  £,  It 

the  E.M.F.  at  middle  of  line  by  £', 

the  E.M.F.,  viz.,  current  at  generator  by  E0)I0\ 


If 


We  have, 


Fig.  84.     Capacity  Shunted  across  Middle  of  Line. 


.   =  I-jbcE' 


E\\  \  (r 


Jbe(r-Jx)         .,    (r-jxy( 

~~ 


or,  expanding, 


[(*  -  bc}  -  (rg+ 


-jx) 


I    (r-jx)(g+jt)-\} 
2  Jf 


110.     ^.)  Z«W  capacity  represented  by  three  condensers^ 
in  the  middle  and  at  the  ends  of  the  line. 
Denoting,  in  Fig.  85, 

the  E.M.F.  and  current  in  receiving  circuit  by  £,  7, 

the  E.M.F.  at  middle  of  line  by  £' ', 


162 


ALTERNATING-CURRENT  PHENOMENA. 


the  current  on  receiving  side  of  line  by  /', 
the  current  on  generator  side  of  line  by  7", 
the  E.M.F.,  viz.,  current  at  generator  by  £0,  f0, 


Iff 


_L  I 


85.     Distributed  Capacity. 


otherwise  retaining  the  same  denotations  as  in  A.), 
We  have, 
7    = 


2"  =  1'  - 


As  will  be  seen,  the  first  terms  in  the  expression   of  E0 
and  of  I0  are  the  same  in  A.)  and  in  B.). 


DISTRIBUTED   CAPACITY.  163 

111.  C.)  Complete  investigation  of  distributed  capacity, 
inductance,  leakage,  and  resistance. 

In  some  cases,  especially  in  very  long  circuits,  as  in 
lines  conveying  alternating  power  currents  at  high  potential 
over  extremely  long  distances  by  overhead  conductors  or  un- 
derground cables,  or  with  very  feeble  currents  at  extremely 
high  frequency,  such  as  telephone  currents,  the  consideration 
of  the  line  resistance  —  which  consumes  E.M.Fs.  in  phase 
with  the  current  —  and  of  the  line  reactance — which  con- 
sumes E.M.Fs.  in  quadrature  with  the  current  —  is  not 
sufficient  for  the  explanation  of  the  phenomena  taking  place 
in  the  line,  but  several  other  factors  have  to  be  taken  into 
account. 

In  long  lines,  especially  at  high  potentials,  the  electro- 
static capacity  of  the  line  is  sufficient  to  consume  noticeable 
currents.  The  charging  current  of  the  line  condenser  is 
proportional  to  the  difference  of  potential,  and  is  one-fourth 
period  ahead  of  the  E.M.F.  Hence,  it  will  either  increase 
or  decrease  the  main  current,  according  to  the  relative  phase 
of  the  main  current  and  the  E.M.F. 

As  a  consequence,  the  current  will  change  in  intensity 
as  well  as  in  phase,  in  the  line  from  point  to  point ;  and  the 
E.M.Fs.  consumed  by  the  resistance  and  inductance  will 
therefore  also  change  in  phase  and  intensity  from  point 
to  point,  being  dependent  upon  the  current. 

Since  no  insulator  has  an  infinite  resistance,  and  as  at 
high  potentials  not  only  leakage,  but  even  direct  escape  of 
electricity  into  the  air,  takes  place  by  "  silent  discharge,"  we 
have  to  recognize  the  existence  of  a  current  approximately 
proportional  and  in  phase  with  the  E.M.F.  of  the  line. 
This  current  represents  consumption  of  energy,  and  is 
therefore  analogous  to  the  E.M.F.  consumed  by  resistance, 
while  the  condenser  current  and  the  E.M.F.  of  inductance 
are  wattless. 

Furthermore,  the  alternate  current  passing  over  the  line 
induces  in  all  neighboring  conductors  secondary  currents, 


164  ALTERNATING-CURRENT  PHENOMENA. 

which  react  upon  the  primary  current,  and  thereby  intro- 
duce E.M.Fs.  of  mutual  inductance  into  the  primary  circuit. 
Mutual  inductance  is  neither  in  phase  nor  in  quadrature 
with  the  current,  and  can  therefore  be  resolved  into  an 
energy  component  of  mutual  inductance  in  phase  with  the 
current,  which  acts  as  an  increase  of  resistance,  and  into 
a  wattless  component  in  quadrature  with  the  current,  which 
decreases  the  self-inductance. 

This  mutual  inductance  is  not  always  negligible,  as, 
for  instance,  its  disturbing  influence  in  telephone  circuits 
shows. 

The  alternating  potential  of  the  line  induces,  by  electro- 
static influence,  electric  charges  in  neighboring  conductors 
outside  of  the  circuit,  which  retain  corresponding  opposite 
charges  on  the  line  wires.  This  electrostatic  influence  re- 
quires the  expenditure  of  a  current  proportional  to  the 
E.M.F.,  and  consisting  of  an  energy  component,  in  phase 
with  the  E.M.F.,  and  a  wattless  component,  in  quadrature 
thereto. 

The  alternating  electromagnetic  field  of  force  set  up  by 
the  line  current  produces  in  some  materials  a  loss  of  energy 
by  magnetic  hysteresis,  or  an  expenditure  of  E.M'.F.  in 
phase  with  the  current,  which  acts  as  an  increase  of  re- 
sistance. This  electromagnetic  hysteretic  loss  may  take 
place  in  the  conductor  proper  if  iron  wires  are  used,  and 
will  then  be  very  serious  at  high  frequencies,  such  as  those 
of  telephone  currents. 

The  effect  of  eddy  currents  has  already  been  referred 
to  under  "mutual  inductance,"  of  which  it  is  an  energy 
component. 

The  alternating  electrostatic  field  of  force  expends 
energy  in  dielectrics  by  what  is  called  dielectric  hysteresis. 
In  concentric  cables,  where  the  electrostatic  gradient  in  the 
dielectric  is  comparatively  large,  the  dielectric  hysteresis 
may  at  high  potentials  consume  considerable  amounts  of 
energy.  The  dielectric  hysteresis  appears  in  the  circuit 


DISTRIBUTED   CAPACITY.  165 

as  consumption  of  a  current,  whose  component  in  phase 
with  the  E.M.F.  is  the  dielectric  energy  current,  which 
may  be  considered  as  the  power  component  of  the  capacity 
current. 

Besides  this,  there  is  the  increase  of  ohmic  resistance 
due  to  unequal  distribution  of  current,  which,  however,  is 
usually  not  large  enough  to  be  noticeable. 

112.  This  gives,  as  the  most  general  case,  and  per  unit 
length  of  line  : 

E.M.Fs.  consumed  in  phase  with  the  current  I,  and  =  rl, 
representing  consumption  of  energy,  and  due  to  : 
Resistance,  and  its  increase  by  unequal  current  distri- 
tribution ;    to   the   energy   component   of   mutual 
inductance;    to  induced  currents ;    to  the  energy 
component   of   self-inductance ;    or  to  electromag- 
netic hysteresis. 
E.M.Fs.  consumed  in  quadrature  with  the  current  I,  and 

=  x  I,  wattless,  and  due  to  : 
Self-inductance,  and  Mutual  inductance. 
Currents  consumed  in  phase  with   the  E.M.F.,   E,   and 
=  gE,  representing  consumption  of  energy,  and 
due  to  : 

Leakage  through   the   insulating  material,  including 
silent    discharge;    energy    component    of    electro- 
static influence ;  energy  component  of  capacity,  or 
of  dielectric  hysteresis. 
Currents  consumed  in  quadrature  to  the  E.M.F.,  E,  and 

=  bE,  being  wattless,  and  due  to  : 
Capacity  and  Electrostatic  influence. 

Hence  we  get  fo'ur  constants  :  — 

Effective  resistance,  r, 
Effective  reactance,  x, 
Effective  conductance,  g, 
Effective  susceptance,  b  —  —  bc, 


1GG     ALTERNATING-CURRENT  PHENOMENA. 

per  unit  length  of  line,  which  represent  the  coefficients,  per 
unit  length  of  line,  of 

E.M.F.  consumed  in  phase  with  current  ; 
E.M.F.  consumed  in  quadrature  with  current  ; 
Current  consumed  in  phase  with  E.M.F.  ; 
Current  consumed  in  quadrature  with  E.M.F. 

113.  This  line  we  may  assume  now  as  feeding  into  a 
receiver  circuit  of  any  description,  and  determine  the  current 
and  E.M.F.  at  any  point  of  the  circuit. 

That  is,  an  E.M.F,  and  current  (differing  in  phase  by  any 
desired  angle)  may  be  given  at  the  terminals  of  receiving  cir- 
cuit.    To  be  determined  are  the  E.M.F.  and  current  at  any 
point  of  the  line  ;  for  instance,  at  the  generator  terminals. 
Or,  Zl=rl—  JXl  ; 

the  impedance  of  receiver  circuit,  or  admittance, 


and  E.M.F.,  E0,  at  generator  terminals  are  given.  Current 
and  E.M.F.  at  any  point  of  circuit  to  be  determined,  etc. 

114.    Counting  now  the  distance,  x,  from  a  point,  0,  of 
the  line  which  has  the  E.M.F., 

•Ei  =  e\  +  Je\i  and  the  current  :  /i  =  i\  +///, 

and  counting  x  positive  in  the  direction  of  rising  energy, 
and  negative  in  the  direction  of  decreasing  energy,  we  have 
at  any  point,  X,  in  the  line  differential,  dx  : 

Leakage  current  :  JEgdx', 
Capacity  current  :   —  j  E  bc  d  x  ; 

hence,  the  total  current  consumed  by  the  line  element,  dx, 
is  dl=  E(g-jbc}d*,  or, 

d-t=E(g-jbc\  (1) 

E.M.F.  consumed  by  resistance,  Ird*\ 
E.M.F.  consumed  by  reactance,    —  j 


DISTRIBUTED    CAPACITY.  107 

hence,  the  total  E.M.F.  consumed  in  the  line  element,  ^/x,  is 
dE    =  I  (r  —  j'x)  </x,   or, 
ffi.   -I(f-jx).  (2) 

These  fundamental  differential  equations : 

*L-E(g-jt,),\  (1) 


(2) 


are  symmetrical  with  respect  to  /  and  E. 
Differentiating  these  equations : 
d*I        dE  , 


and  substituting  (1)  and  (2)  in  (3),  we  get : 

(4) 


(5) 
the  differential  equations  of  E  and  L 

115.  These  differential  equations  are  identical,  and  con- 
sequently I  and  E  are  functions  differing  by  their  limiting 
conditions  only. 

These  equations,  (4)  and  (5),  are  of  the  form  : 

(6) 


and  are  integrated  by 

W  =  tf  6rx, 

where  e  is  the  basis  of   natural  logarithms  ;  for,  differen- 
tiating this,  we  get, 


168  ALTERNATING-CURRENT  PHENOMENA. 

hence,  z>2  =  (g  —  j  bc)  (r  —  jx)  ; 


(7) 


or,  v  =  ±  V  (g  -  Jbe)  (r  —  joe)  \ 

hence,  the  general  integral  is  : 

tr*.«e+«-M«r««  (8) 

where  a  and  b  are  the  two  constants  of  integration  ; 
Substituting 

r-«-/0  (9) 

into  (7),  we  have, 

(a  -JP)*  =  (g  -  jbc)  (r  -  jx)  ; 
or, 


therefore,        _  f 

);-'  (10) 


Vl/2          6  -   e 


/3=  Vl/2 
substituting  (9)  into  (8)  : 


=  a-cax  (cos/3x  —  /sin^Sx)  +  ^c~ax  (cos/3x  +y  sin/3x)  ; 
«/  =  (a£«x  +  /5>e~ax)  cos)8x  —  y(aeax  —  ^«-ax)  sin  /3x  (12) 

which  is  the  general  solution  of  differential  equations  (4) 
and  (5) 

Differentiating  (8)  gives  : 


hence,  substituting,  (9)  : 
(a  —JP)  {(a 


x}.  (13) 

Substituting  now  /  for  w,  and  substituting  (13)  in  (1), 
and  writing, 


DISTRIBUTED    CAPACITY. 


169 


we  get, 


/•  \(  Jfax.  _i_    > 

?e-«)cosj8x-y(y 
?«-«)cos/8x-y(y 

-• 

*   —                •/_>    <  \                ' 
a  —  7/5 

sin  /2x}  ; 

'**     1     K^"    i 

S  —  J^c 
sin  ySxf  ; 

where  ^4  and  ^  are  the  constants  of  integration. 
Transformed,  we  get, 

/= J  Aea*   (cos    )8x  —  j  sin    0x)  +  Bf.~™ 

a  — JP   (  ' 

(cos  /?x  +/  sin  /8x)  > 
1 


^4eax    (cos    /8x  —  y    sin 


^-. 

(cos  /3x  +y  sin  y8x) 

Thus  the  waves  consist  of  two  components,  one,  with 
factor  ^eax,  increasing  in  amplitude  toward  the  generator, 
the  other,  with  factor  ^e-ax,  decreasing  toward  the  genera- 
tor. The  latter  may  be  considered  as  a  reflected  wave. 

At  the  point  x  =  0. 


a-j/3 
A-B 


n 

Thus  m  (cos  to  —  j  sin  G)  =  -— 

and, 

m  =  amplitude. 

w  =  angle  of  reflection. 

These  are  the  general  integral  equations  of  the  problem. 


116.    If  — 

/!  =  /!  +  ///  is  the  current 
{  is  the  E.M.F. 


at  point,  x 


(15) 


170  ALTERNATING-CURRENT  PHENOMENA. 

by  substituting  (15)  in  (14),  we  get : 
2  A  =  {(a  t\  +  ft  //)  +  (gev  +  bc  ^') 

(16) 


2  B  =  {(a  /!  +  /?  //)  -  (ge,  +  /;c  ,/)} 
+  /{(«//-  0/0  -(^I'-^ 
a  and  ft  being  determined  by  equations  (11). 

117.    H  Z  —  R  —  j  X  is  the  impedance  of  the  receiver 
circuit,  E0  =  e0  +  j >0'  is  the  E.M.F.  at  dynamo  terminals 
(17),  and  /  =  length  of  line,  we  get 
at 


hence 


g  —  jbc 
or 


a-;  ft 
At  X  =  /, 


E0 


sin/?/}.  (19) 


Equations  (18)  and  (19)  determine  the  constants  A  and  B, 
which,  substituted  in  (14),  give  the  final  integral  equations. 

The  length,  X0  =  2  TT  /  ft  is  a  complete  wave  length  (20), 
,vhich  means,  that  in  the  distance  2  IT  /  ft  the  phases  of  the 
components  of  current  and  E.M.F.  repeat,  and  that  in  half 
this  distance,  they  are  just  opposite. 

Hence  the  remarkable  condition  exists  that,  in  a  very 
long  line,  at  different  points  the  currents  at  the  same  time 
flow  in  opposite  directions,  and  the  E.M.Fs.  are  opposite. 

118.    The  difference  of  phase,  w,  between  current,  /,  and 
E.M.F.,  Ey  at  any  point,  x,  of  the  line,  is  determined  by 


DISTRIBUTED    CAPACITY.  171 


the  equation, 

Z?(cos«+/sin£)  =y,  :  \j  JsTI71 

where  Z>  is  a  constant. 

Hence,  w  varies  from  point  to  point,  oscillating  around  a 
medium  position,  wx,  which  it  approaches  at  infinity. 

This  difference  of  phase,  C>x,  towards  which  current  and 
E.M.F.  tend  at  infinity,  is  determined  by  the  expression, 


^(cos      .    ..  ,         (/ 

or,  substituting  for  E  and  /their  values,  and  since  e~a*  =  0, 
and  A  eax  (cos  ft  x  —  j  sin  ft  x),  cancels,  and 

D  (cos  tow  +/sin  oioc)  =  — 2-p- 


hence,  tan  ^  =  ~a°  c  +  ^    •  (21) 

This  angle,  Stx,  =  0  ;  that  is,  current  and  E.M.F.  come 
more  and  more  in  phase  with  each  other,  when 

abc  —  fig  —  0 ;  that  is, 

a  -T-  ft  —  g  -r-  bc ,  or, 

2a/3       !*2^*/  5 
substituting  (10),  gives, 


hence,  expanding,         r  -4-  ^  =  ^  -f-  ^c  ;  (22) 

that  is,  tJie  ratio  of  resistance  to  inductance  equals  the  ratio 
of  leakage  to  capacity. 

This  angle,  wx,  =  45°  ;  that  is,  current  and  E.M.F.  differ 
by  £th  period,  if  —  a  bc  +  fig  =  a.g  +  pbc  ;  or, 


which  gives  :  rg  +  x  bc  =  0.  (23) 


172 


ALTERNA TING-CURRENT  PHENOMENA. 


That  is,  two  of  the  four  line  constants  must  be  zero;  cither 
g  and  x,  or  g  and  bc. 

The  case  where  g  =  0  =  x,  that  is  a  line  having  only 
resistance  and  distributed  capacity,  but  no  self-induction  is 
approximately  realized  in  concentric  or  multiple  conductor 
cables,  and  in  these  the  phase  angle  tends  towards  45°  lead 
for  infinite  length. 

119.  As  an  instance,  in  Fig.  86  a  line  diagram  is  shown, 
with  the  distances  from  the  receiver  end  as  abscissae. 
The  diagram  represents  one  and  one-half  complete  waves, 
and  gives  total  effective  current,  total  E.M.F.,  and  differ- 


<± 

"o^ 

+  30 

'»sr 
I 

\ 

OLT» 
.0,000 

»20 

i 

\ 

8()0{ 

1 
1 

\ 

/ 

• 

*\ 

£ 

j 

ja 

u 

i 

\ 

X 

• 

•*> 

V 

u 

( 

\ 

^    - 

+'' 

/ 

-20 

\ 

/ 

/ 

-30 

/ 

"*"•> 

/ 

•  

-40. 

: 

.„, 

•us 
Kl 

s 

I 
-50 

/ 

/.o 

7,0 

/ 

/ 

z;o 

/ 

/ 

p. 

„. 

,  — 

•j^, 

/ 

u-j 

c 

/ 

'' 

/ 

.- 

S 

«.ooo 

/ 

?    ' 

N 

/ 

/ 

'  000 

/ 

\ 

^~ 

/ 

V 

100 

0,00, 

X 

/ 

N>, 

.s 

x  =  ' 

60 

9  000 

/ 
/ 

g=i 

bc= 

XI 
'X| 

rj-4 

»0 

«.ooo 

\ 

\ 

/ 

4,000 

\, 

-"* 

JO 

J.OOO 

0 

i 

3 

- 

J 
\ 

L 

5 
1 

L 

i 

Fig.  86. 


DISTRIBUTED    CAPACITY.  173 

ence  of  phase  between  both  as  function  of  the  distance  from 
receiver  circuit ;  under  the  conditions, 

E.M.F.  at  receiving  end,  10,000  volts;  hence,  Ev  =el  =  10,000; 
current  at  receiving  end,  65  amperes,  with  a  power  factor 
of  .385. 

that  is,  /  =  t\  +  j  //  =  25  +  60  j ; 

line  constants  per  unit  length, 

r  =  1,  g  =  2  X  10-5, 

hence, 

a  =  4.95  x  10-3,  ] 
13  =  28.36  x  10 -3,  j- 


length  of  line  corresponding  to 
one  complete  period  of  the  wave 


x0  =  L  =  —  =  221.5  = 

(^     of  propagation. 
A  =  1.012  -  1.206  y  ) 
B  =    .812  +    .794  /  j 

These  values,  substituted,  give, 

/=     {£«x  (47.3  cos  /?x  +  27.4  sin  fix)  —  e-«* 

(22.3  cos  ftx  +  32.6  sin  fix)} 
+  y  (e«x  (27.4  cos  ftx  —  47.3  sin  ftx)  +  €-«x 

(32.6  cos  y3x  —  22.3  sin  /3x)}; 
E  =     {eox  (6450  cos  /3x  +  4410  sin  j8x)  +  c-ax 

(3530  cos  fix  +  4410  sin  /?x)} 
+  y  (eox  (4410  cos  /3x  —  6450  sin  £x)  —  e~ax 
(4410  cos  ft  x-  3530  sin  /3x)}; 

tan  5,  =  ~  °-ljc  +  PS  =  _  .073,          JJ«  =  -  4.2°. 

120.  As  a  further  instance  are  shown  the  characteristic 
curves  of  a  transmission  line  of  the  relative  constants, 

r\x\g>.b  =  %  :  32  :  1.25  X  10  ~4 :  25  X  10  ~4,  and  e 
=  25,000,  i  =  200  at  the  receiving  circuit,  for  the  con- 
ditions, 

a,  non-inductive  load  in  the  receiving  circuit,  Fig.  87. 


174 


ALTERNATING-CURRENT  PHENOMENA. 


b,  wattless  receiving  circuit  of  90°  lag,  Fig.  88. 

c,  wattless  receiving  circuit  of  90°  lead,  Fig.  89. 
These  curves  are  determined  graphically  by  constructing 

the  topographic  circuit  characteristics  in  polar  coordinates 
as  explained  in  Chapter  VI.,  paragraphs  36  and  37,  and  de- 
riving corresponding  values  of  current,  potential  difference 
and  phase  angle  therefrom. 

As  seen  from  these  diagrams,  for  wattless  receiving  cir- 
cuit, current  and  E.M.F.  oscillate  in  intensity  inversely  to 


ZJ 

7 

6sa 


7 


rig.  87. 


DISTRIBUTED    CAPACITY. 


175 


each  other,  with  an  amplitude  of  oscillation  gradually  de- 
creasing when  passing  from  the  receiving  circuit  towards 
the  generator,  while  the  phase  angle  between  current  and 
E.M.F.  oscillates  between  lag  and  lead  with  decreasing  am- 
plitude. Approximately  maxima  and  minima  of  current  co- 
incide with  minima  and  maxima  of  E.M.F.  and  zero  phase 
angles. 


\ 


V 


Fig.  88. 


176 


AL  TERNA  TING-CURRENT  PHENOMENA. 


For  such  graphical  constructions,  polar  coordinate  paper 
and  two  angles  a  and  8  are  desirable,  the  angle  a  being  the 

angle  between  current  and  change  of  E.M.F.,  tan  a  =  -  =  4, 
and  the  angle  8  the  angle  between  E.M.F.  and  change  of 

current,  tan  8  =  -  =  20  in  above  instance. 
g 


\ 


Fig.  89. 


DISTRIBUTED    CAPACITY. 


177 


With  non-inductive  load,  Fig.  87,  these  oscillations  of 
intensity  have  almost  disappeared,  and  only  traces  of  them 
are  noticeable  in  the  fluctuations  of  the  phase  angle  and  the 
relative  values  of  current  and  E.M.F.  along  the  line. 

Towards  the  generator  end  of  the  line,  that  is  towards 
rising  power,  the  curves  can  be  extended  indefinitely,  ap- 
proaching more  and  more  the  conditions  of  non-inductive 
circuit,  towards  decreasing  power,  however,  all  curves  ulti- 
mately reach  the  conditions  of  a  wattless  receiving  circuit, 
as  Figs.  88  and  89,  at  the  point  where  the  total  energy  in- 


t 


a  +120 


ISSION    LINE 


V 


Fig.  90. 

put  into  the  line  has  been  consumed  therein,  and  at  this 
point  the  two  curves  for  lead  and  for  lag  join  each  other  as 
shown  in  Fig.  90,  the  one  being  a  prolongation  of  the  other, 
and  the  flow  of  power  in  the  line  reverses.  Thus  in  Fig.  90 
power  flows  from  both  sides  of  the  line  towards  the  point  of 
zero  power  marked  by  0,  where  current  and  E.M.F.  are  in 
quadrature  with  each  other,  the  current  being  leading  with 
regard  to  the  flow  of  power  from  the  left,  and  lagging  with 
regard  to  the  flow  of  power  from  the  right  side  of  the 
diagram. 


178  DISTRIBUTED    CAPACITY. 

121.     The  following  are  some  particular  cases : 
A.)    Open  circuit  at  end  of  lines : 
x  =  0 :  /!  =  0. 

hence, 

E  = i-r—  ^{(eax  +  e-ax)  cos/3x  —  y(cax  —  c-ax)sin/3x}; 


.£?.)    Line  grounded  at  end: 


A  —  (a/\  -J-  /?//)  +/  (a//  —  ^zi)  =  -? 
-^-T-^{(eax  —  c-ax)  cos/?x  —  >(eax  +  c~ax)  sin)8x}; 


(T.)    Infinitely  long  conductor  : 

Replacing  x  by  —  x,  that  is,  counting  the  distance  posi- 
tive in  the  direction  of  decreasing  energy,  we  have, 

x  =  oo  :  7=  0,  E  =  0; 
hence 


and 


I  =  —  -  —  ^£-°x(cos/Sx  +ysin/3x), 

' 


revolving  decay  of  the  electric  wave,  that  is  the  reflected 
wave  does  not  exist. 

The  total  impedance  of  the  infinitely  long  conductor  is 


(q-yff)  (g+M 


+  b?  g*  +  b* 


ALTERNATING-CURRENT  PHENOMENA.  179 

The  infinitely  long  conductor  acts  like  an  impedance 

7  _  °-K  +  P  ?>c  _    •  fig  —  Q-bc 

f*+v      g*  +  K' 

that  is,  like  a  resistance 


combined  with  a  reactance 


We  thus  get  the  difference  of  phase  between   E.M.F. 
and  current, 


which  is  constant  at  all  points  of  the  line. 
If  g  =  0,  x  =  0,  we  have, 


hence, 

tan  to  =  1,  or, 

£  =  45°  ; 

that  is,  current  and  E.M.F.  differ  by  £th  period. 
D.)    Generator  feeding  into  closed  circuit  : 
Let  x  =  0  be  the  center  of  cable  ;  then, 

hence  :    E  —  0  at  x  =  0  ; 


which  equations  are  the  same  as  in  B,  where  the  line  is 
grounded  at  x  =  0. 

E.)    Let  the  length  of  a  line  be  one-quarter  wave  length; 


and  assume  the  resistance  r  and  conductance  g  as  negligible 


180  AL  TERN  A  TING-CURRENT  PHENOMENA. 

compared  with  x  and  bc. 

r=0=g 
These  values  substituted  in  (11)  give 

a=0. 

(3=  V^ 

Let  the   E.M.F.  at   the  receiving  end  of   the  line  be 
assumed  zero  vector 

£l  =  ei  =  E.M.F.  and 

fi  —  i'i  + ji\.  —  current  at  end  of  line     x  =  0 
£0  =  E.M.F.  and 

S0  =  current  at  beginning  of  line 


Substituting  in  (16)  these  values  of  El  and  7:  and  also  r  =  0 
=  g,  we  have 


From  these  equations  it  follows  that 


which  values,  together  with  the  foregoing  values  of  Ev  Iv  r, 
g,  a,  and  /8,  substituted  in  (14)  reduce  these  equations  to 


—  j  (i\  +jiC)  \~r  s^ 


ALTERNATING-CURRENT  TRANSFORMER,          181 

Then  at     x 


Hence  also 


•£"„  and  70  are  both  in  quadrature  ahead  of  <?x  and  7j 
respectively. 

Il  =  EQ  y  —  =  constant,  if   7f0  =  constant.     That  is,  at 

constant  impressed  E.M.F.  E&  the  current  7X  in  the  receiv- 
ing circuit  of  a  line  of  one-quarter  wave  length  is  constant, 
and  inversely  (constant  potential  —  constant  current  trans- 
formation by  inductive  line).  In  this  case,  the  current  70  at 
the  beginning  of  the  line  is  proportional  to  the  load  el  at  the 
end  of  the  line. 

If  XQ  =  lx  =  total  reactance, 

b0  =  lbc  =  total  susceptance  of  line,  then 

*<A>  =  4- 

Instance*  =  4,  bc  =  20  X  10  ~5,  E0  =  10,000  V.  Hence 
/  =  55.5,  *0  =  222,  b0  =  .0111,  7j  =  70.7,  70  =  .00707  e. 

122.  An  interesting  application  of  this  method  is  the 
determination  of  the  natural  period  of  a  transmission  line  ; 
that  is  the  frequency  at  which  such  a  line  discharges  an 
accumulated  charge  of  atmospheric  electricity  (lightning), 
or  oscillates  at  a  sudden  change  of  load,  as  a  break  of  cir- 
cuit. 


182  ALTERNATING-CURRENT  PHENOMENA. 

The  discharge  of  a  condenser  through  a  circuit  contain- 
ing self-induction  and  resistance  is  oscillating  (provided  that 
the  resistance  does  not  exceed  a  certain  critical  value  de- 
pending upon  the  capacity  and  the  self-induction).  That  is, 
the  discharge  current  alternates  with  constantly  decreasing 
intensity.  The  frequency  of  this  oscillating  discharge  de- 
pends upon  the  capacity,  C,  and  the  self-induction,  L,  of  the 
circuit,  and  to  a  much  lesser  extent  upon  the  resistance,  so 
that  if  the  resistance  of  the  circuit  is  not  excessive  the  fre- 
quency of  oscillation  can,  by  neglecting  the  resistance,  be 
expressed  with  fair,  or  even  close,  approximation  by  the 
formula  - 


An  electric  transmission  line  represents  a  capacity  as  well 
as  a  self-induction  ;  and  thus  when  charged  to  a  certain 
potential,  for  instance,  by  atmospheric  electricity,  as  by  in- 
duction from  a  thunder-cloud  passing  over  or  near  the  line, 
the  transmission  line  discharges  by  an  oscillating  current. 

Such  a  transmission  line  differs,  however,  from  an  ordi- 
nary condenser,  in  that  with  the  former  the  capacity  and 
the  self-induction  are  distributed  along  the  circuit. 

In  determining  the  frequency  of  the  oscillating  discharge 
of  such  a  transmission  line,  a  sufficiently  close  approximation 
is  obtained  by  neglecting  the  resistance  of  the  line,  which, 
at  the  relatively  high  frequency  of  oscillating  discharges, 
is  small  compared  with  the  reactance.  This  assumption 
means  that  the  dying  out  of  the  discharge  current  through 
the  influence  of  the  resistance  of  the  circuit  is  neglected, 
and  the  current  assumed  as  an  alternating  current  of  ap- 
proximately the  same  frequency  and  the  same  intensity  as 
the  initial  waves  of  the  oscillating  discharge  current.  By 
this  means  the  problem  is  essentially  simplified. 

Let  /  =  total  length  of  a  transmission  line, 
r  =  resistance  per  unit  length, 
x  =  reactance  per  unit  length  =  2  ?r  NL. 


DISTRIBUTED   CAPACITY.  183 

where  L  =  coefficient  of   self-induction  or  inductance  per  unit 

length ; 

g  =  conductance  from  line  to  return  (leakage  and  dis- 
charge into  the  air)  per  unit  length ; 
b  =  capacity  susceptance  per  unit  length  =  2  TT  NC 
where    C  =  capacity  per  unit  length. 

x  =  the  distance  from  the  beginning  of  the  line, 

We  have  then  the  equations  : 
The  E.M.F., 


(^eax    _   ^e-ax)      CQS      £x    _j     (4€ 

g  —  jb  I       +  ^e~ax)  sin  /3x 
the  current, 

1        ^  (Aea*  +  ^e~ax)    COS    /3x  — y    (^4e 


where, 


,(14.) 


(r1  +  ^c2)  +  (^r  - 

'  (11.) 


c    =  base  of  the  natural  logarithms,  and  A  and  B  integration 
constants. 

Neglecting  the  line  resistance,  r  =  0,  and  the  conduc- 
tance (leakage,  etc.),  g=0,  gives, 


These  values  substituted  in  (14)  give, 
J- 


=  J-\(A  -     B}  cos   ^fbx^  -j  (A  +  H)  sin 


/  =  -4=  J  (^  +  -ff)  cos  V^x  —  y  (<4  -  B)  sin 

;  J 


184  ALTERNATING-CURRENT  PHENOMENA. 

If  the  discharge  takes  place  at  the  point :  x  =  0,  that  is, 
if  the  distance  is  counted  from  the  discharge  point  to  the 
end  of  the  line  ;  x  =  /,  hence  : 

At  x  =  0,  E  =  0, 
Atx=/,     7=0. 

Substituting  these  values  in  (25)  gives, 

For  x  =  0, 

^-7^  =  0        A  =  B 

which  reduces  these  equations  to, 

E  =  — —  sin  Nbx  x 

b  \ 

7=  -^4^=  cos  V&t:  x 

VA*  I 

and  at  x  =  0, 


At  x  =  /,  /  =  0,  thus,  substituted  in  (26), 

cos  V^/  =  0  (28.) 

hence  : 

V^/^2**1)",  1  =  0,1,  2,...  (29.) 

that  is,  *Jbx  I  is  an  odd  multiple  of  ^  •     And  at  x  =  /, 

2t 

O     A 


Substituting  in  (29)  the  values, 

we  have, 

hence, 

^=M  +  l  (31.) 

4/VCZ 


DISTRIBUTED   CAPACITY.  185 

the  frequency  of  the  oscillating  discharge, 
where  k  =  0,  1,  2.  .  .  . 

That  is,  the  oscillating  discharge  of  a  transmission  line 
of  distributed  capacity  does  not  occur  at  one  definite  fre- 
quency (as  that  of  a  condenser),  but  the  line  can  discharge 
at  any  one  of  an  infinite  number  of  frequencies,  which  are 
the  odd  multiples  of  the  fundamental  discharge  frequency, 

*-I7^z  (32'> 

Since 

C0  =  1C  =  total  capacity  of  transmission  line,  ) 

L0  =  IL  =  total  self-inductance  of  transmission  line,  J  ^     '' 

we  have, 

2,£  +  1 

-=  the  frequency  of  oscillation,  (34.) 


or  natural  period  of  the  line,  and 

NI  —  -  -  the  fundamental, 
- 


or  lowest  natural  period  of  the  line. 
From  (30),  (33),  and  (34), 

b  =  2irNC= 2/        \T0  (36-) 

and  from  (29), 

V^  =  (2^2f/)7r-  <37') 

These  substituted  in  (26)  give, 

f-  (38.) 
4/7  (2£  +  l)7rx 

/=(2TTi)-^cosL^H 

The  oscillating  discharge  of  a  line  can  thus  follow  any  of 
the  forms  given  by  making  k  —  0,  1,  2,  3  .  .  .in  equation 
(38). 

Reduced  from  symbolic  representation  to  absolute  values 


186  ALTERNATING-CURRENT  PHENOMENA. 

by  multiplying  E  with  cos  2  *  Nt  and  /  with  sin  2  TT  A7/  and 
omitting  j,  and  substituting  A7"  from  equation  (34),  we  have, 

(2£+l)7rx 

—      sin  —     JT—  -  —  -cos 
2/ 


where  ^4  is  an  integration  constant,  depending  upon  the 
initial  distribution  of  voltage,  before  the  discharge,  and 
/  =  time  after  discharge. 

123.    The  fundamental  discharge  wave  is  thus,  for  k  =  0, 


47.       Lo        .       .        7TX  7T/ 

-^  A  sin  7^ 
C0  2/ 


.  o        .       .        7TX 

=  —  \  -^  A  sin  7^—  cos 

TT  V 


4  /  -    _,  7T  X  7T/ 

fi  =  —  A  cos  7n-  sin  - 


With  this  wave  the  current  is  a  maximum  at  the  begin- 
ning of  the  line :  x  =  0,  and  gradually  decreases  to  zero  at 
the  end  of  the  line  :  x  =  /. 

The  voltage  is  zero  at  the  beginning  of  the  line,  and 
rises  to  a  maximum  at  the  end  of  the  line. 

Thus  the  relative  intensities  of  current  and  potential 
along  the  line  are  as  represented  by  Fig.  91,  where  the  cur- 
is  shown  as  /,  the  potential  as  E. 

The  next  higher  discharge  frequency,  for  :  k  —  1,  gives  : 

47.  [Ln    .    .    3v_ 

(41.) 

4/  "  "    -  ' 

/,  =  o-  A  cos 


n  7 
27 


DISTRIBUTED   CAPACITY. 


187 


Here  the  current  is  again  a  maximum  at  the  beginning 
of   the  line :    x  =  0,  and  gradually  decreases,  but  reaches 

zero  at  one-third  of  the  line  :   x  =  _,  then  increases  again,  in 

o 


Fig. 


H----0 


Fig. 


\ 


\ 


\ 


\1 


Figs.  91-93. 


188  ALTERNATING   CURRENT-PHENOMENA. 

the  opposite  direction,  reaches  a  second  but  opposite  maxi- 

2/ 

mum  at  two-thirds  of  the  line  :    x  =  ^—  ,  and  decreases  to 

o 

zero  at  the  end  of  the  line.  There  is  thus  a  nodal  point  of 
current  at  one-third  of  the  line. 

The  E.M.F.  is  zero  at  the  beginning  of  the  line  :    x  =  0, 

rises  to  a  maximum  at  one-third    of   the  line  :    x  =  -  ,  de- 

2/        3 
creases  to  zero  at  two-thirds  of  the  line  :    x  =  IT  >  and  rises 

again  to  a  second  but  opposite  maximum  at  the  end  of  the 
line:  x  =  /.  The  E.M.F.  thus  has  a  nodal  point  at  two- 
thirds  of  the  line. 

The  discharge  waves  :  k  =  1,  are  shown  in  Fig.  92, 
those  with  k  =  2,  with  two  nodal  points,  in  Fig.  93. 

Thus  k  is  the  number  of  nodal  points  or  zero  points  of 
current  and  of  E.M.F.  existing  in  the  line  (not  counting 
zero  points  at  the  ends  of  the  line,  which  of  course  are  not 
nodes). 

In  case  of  a  lightning  discharge  the  capacity  C0  is  the 
capacity  of  the  line  against  ground,  and  thus  has  no  direct 
relation  to  the  capacity  of  the  line  conductor  against  its 
return.  The  same  applies  to  the  inductance  L0. 

If  d  =  diameter  of  line  conductor, 

D  =  distance  of  conductor  above  ground, 
and  /  =  length  of  conductor, 

the  capacity  is, 


1.11  x  10-6/     ,. 

~ 


the  self-inductance, 


The  fundamental  frequency  of  oscillation  is  thus,  by 
substituting  (42)  in  (35), 


DISTRIBUTED   CAPACITY.  189 

That  is,  the  frequency  of  oscillation  of  a  line  discharging 
to  ground  is  independent  of  the  size  of  line  wire  and  its 
distance  from  the  ground,  and  merely  depends  upon  the 
length  /  of  the  line,  being  inversely  proportional  thereto. 

We  thus  get  the  numerical  values, 

Length  of  line 

10       20       30       40        50     60      80     100  miles. 
=     1.6      3.2      4.8      6.4         8     9.6  12.8  16  x  106  cm.. 

hence  frequency, 

N-i  =  4680  2340  1560  1170  937.5  780    585  475  cycles-.. 

As  seen,  these  frequencies  are  comparatively  low,  and 
especially  with  very  long  lines  almost  approach  alternator 
frequencies. 

The  higher  harmonics  of  the  oscillation  are  the  odd! 
multiples  of  these  frequencies. 

Obviously  all  these  waves  of  different  frequencies  repre- 
sented in  equation  (39)  can  occur  simultaneously  in  the 
oscillating  discharge  of  a  transmission  line,  and  in  general 
the  oscillating  discharge  of  a  transmission  line  is  thus  of 
the  form, 

(by  substituting:  ak  =         *     j 


where  a^  as  ay  .  .  .  are  constants  depending  upon  the 
initial  distribution  of  potential  in  the  transmission  line,  at 
the  moment  of  discharge,  or  at  /  =  0,  and  calculated  there- 
from. 


190  AL  TERN  A  TING-CURRENT  PHENOMENA . 

124.  As  an  instance  the  following  discharge  equation 
of  a  line  charged  to  a  uniform  potential  e  over,  its  entire 
length,  and  then  discharging  at  x  =  0,  has  been  calculated. 

The  harmonics  are  determined  up  to  the  11 — that  is,  av 

•a&  #5>  av  a9>  an- 

These  six  unknown  quantities  require  six  equations,  which 

/     2/    3/  4/   5/    6/ 
are  given  by  assuming  E  =  e  for  x  =  g,  _,_,_,_,_. 

At  /  =  0,  E  =  e,  equation  (44)  assumes  the  form 

4  /    HT   (        .    TTX   ,         .    3  TTX 
e  =  —  V  £?  j  «i sm  27  +  *3  sm~27  +  '    '    '    '   +  *u 

(45.) 

/     2/  6/ 

Substituting  herein  for  x  the  values  :  - ,  — ,  .  .  .  — 

gives  six  equations  for  the  determination  of  av  <73  .  .  .  an. 
These  equations  solved  give, 


E  =  e  (1.26  sin  w  cos  $  +  .40  sin  3  w  cos  3  <f»  +  .22  sin  ^ 
5  w  cos  5  <^  +  .12  sin  7  o>  cos  7  <£  +  .07  sin  9  co 
cos  9  ^  +  .02  sin  11  o>  cos  11  ^> 

5 

L0 

cos  5  <o  sin  5  <^  +  .12  cos  7  o>  sin  7  <£  +  .07  cos 
9  to  sin  9  <£  +  .02  cos  11  o>  sin  11  </> 


7  =  e i/5 (1.26  cos  o>  sin  <£  +  .40  cos  3  w  sin  3  <£  +  .22 

V    7rt 


,(46.) 


where, 

"-57  1 

„  r<47') 


Instance,     . 

Length  of  line,  /  =  25  miles  =  4  x  106  cm. 
Size  of  wire :  No.  000  B.  &  S.  G.,  thus :  d  =  1  cm. 
Height  above  ground :  D  —  18  feet  =  550  cm. 
Let  e  =  25,000  volts  =  potential  of  line  in  the  moment  of 
-discharge. 


DISTRIBUTED  CAPACITY.  191 

We  then  have, 

E  =  31,500  sin  w  cos  <fr  +  10,000  sin  3  <o  cos  3  <£  +  5500  sin 

5  o>  cos  5  <J>  +  3000  sin  7  o>  cos  7  <£  -j-  1750  sin  9  o>  cos 

9  <£  +  500  sin  11  w  cos  11  <£. 
/=  61.7  cos  w  sin  <£  +  19.6  cos  3  o>  sin  3  <£  +  10.8  cos  5  «  sin 

5  <£  +  5.9  cos  7  CD  sin  7  </>  +  3.4  cos  9  to  sin  9  <£  +  1.0 

cos  11  <o  sin  11  <J>. 

<o=    .39 .r  10 -6 
</>  =  1.18/  10+4 

A  simple  harmonic  oscillation  as  a  line  discharge  would 
require  a  sinoidal  distribution  of  potential  on  the  trans- 
mission line  at  the  instant  of  discharge,  which  is  not  proba- 
ble, so  that  probably  all  lightning  discharges  of  transmission 
lines  or  oscillations  produced  by  sudden  changes  of  circuit 
conditions  are  complex  waves  of  many  harmonics,  which  in 
their  relative  magnitude  depend  upon  the  initial  charge  and 
its  distribution  —  that  is,  in  the  case  of  the  lightning  dis- 
charge, upon  the  atmospheric  electrostatic  field  of  force. 

The  fundamental  frequency  of  the  oscillating  discharge 
of  a  transmission  line  is  relatively  low,  and  of  not  much 
higher  magnitude  than  frequencies  in  commercial  use  in 
alternating  current  circuits.  Obviously,  the  more  nearly 
sinusoidal  the  distribution  of  potential  before  the  discharge, 
the  more  the  low  harmonics  predominate,  while  a  very  un- 
equal distribution  of  potential,  that  is  a  very  rapid  change 
along  the  line,  as  caused  for  instance  by  a  sudden  short 
circuit  rupturing  itself  instantly,  causes  the  higher  harmo- 
nics to  predominate,  which  as  a  rule  are  more  liable  to  cause 
excessive  rises  of  voltage  by  resonance. 

125.  As  has  been  shown,  the  electric  distribution  in  a 
transmission  line  containing  distributed  capacity,  self-induc- 
tion, etc.,  can  be  represented  either  by  a  polar  diagram 
with  the  phase  as  amplitude,  and  the  intensity  as  radius 
vector,  as  in  Fig.  34,  or  by  a  rectangular  diagram  with  the 


192  ALTERNATING-CURRENT  PHENOMENA. 

distance  as  abscissae,  and  the  intensity  as  ordinate,  as  in 
Fig.  35  and  in  the  preceding  paragraphs. 

In  the  former  case,  the  consecutive  points  of  the  circuit 
characteristic  refer  to  consecutive  points  along  the  trans- 
mission line,  and  thus  to  give  a  complete  representation  of 
the  phenomenon,  should  not  be  plotted  in  one  plane  but  in 
front  of  each  other  by  their  distance  along  the  transmission 
line.  That  is,  if  0,  1,  2,  etc.,  are  the  polar  vectors  in  Fig. 
34,  corresponding  to  equi-distant  points  of  the  transmission 
line,  1  should  be  in  a  plane  vertically  in  front  of  the  plane 
of  0,  2  by  the  same  distance  in  front  of  1,  etc. 

In  Fig.  35  the  consecutive  points  of  the  circuit  charac- 
teristic represent  vectors  of  different  phase,  and  thus  should 
be  rotated  out  of  the  plane  around  the  zero  axis  by  the 
angles  of  phase  difference,  and  then  give  a  length  view  of 
the  same  space  diagram,  of  which  Fig.  34  gives  a  view  along 
the  axis. 

Thus,  the  electric  distribution  in  a  transmission  line  can 
be  represented  completely  only  by  a  space  diagram,  and  as 
complete  circuit  characteristic  we  get  for  each  of  the  lines 
a  screw  shaped  space  curve,  of  which  the  distance  along  the 
axis  of  the  screw  represents  the  distance  along  the  transmis- 
sion line,  and  the  distance  of  each  point  from  the  axis  rep- 
resents by  its  direction  the  phase,  and  by  its  length  the 
intensity. 

Hence  the  electric  distribution  in  a  transmission  line 
leads  to  a  space  problem  of  which  Figs.  34  and  35  are  par- 
tial views.  The  single-phase  line  is  represented  by  a  double 
screw,  the  three-phase  line  by  a  triple  screw,  and  the  quarter- 
phase  four-wire  line  by  a  quadruple  screw.  In  the  symbolic 
expression  of  the  electric  distribution  in  the  transmission 
line,  the  real  part  of  the  symbolic  equation  represents  a  pro- 
jection on  a  plane  passing  through  the  axis  of  the  screw, 
and  the  imaginary  part  a  projection  on  a  plane  perpendicular 
to  the  first,  and  also  passing  through  the  axis  of  the  screw. 


ALTERNATING-CURRENT  TRANSFORMER.     193 


CHAPTER    XIV. 

THE    ALTERNATING-CURRENT    TRANSFORMER. 

126.  The  simplest  alternating-current  apparatus  is  the 
transformer.     It  consists  of  a  magnetic  circuit  interlinked 
with  two  electric  circuits,  a  primary  and  a  secondary.     The 
primary  circuit  is  excited  by  an  impressed  E.M.F.,  while  in 
the  secondary  circuit  an  E.M.F.  is  induced.     Thus,  in  the 
primary  circuit  power  is  consumed,  and  in  the  secondary 
a  corresponding  amount  of  power  is  produced. 

Since  the  same  magnetic  circuit  is  interlinked  with  both 
electric  circuits,  the  E.M.F.  induced  per  turn  must  be  the 
same  in  the  secondary  as  in  the  primary  circuit  ;  hence, 
the  primary  induced  E.M.F.  being  approximately  equal  to 
the  impressed  E.M.F.,  the  E.M.Fs.  at  primary  and  at  sec- 
ondary terminals  have  approximately  the  ratio  of  their 
respective  turns.  Since  the  power  produced  in  the  second- 
ary is  approximately  the  same  as  that  consumed  in  the 
primary,  the  primary  and  secondary  currents  are  approxi- 
mately in  inverse  ratio  to  the  turns. 

127.  Besides  the  magnetic   flux  interlinked  with  both 
electric  circuits  —  which  flux,  in  a  closed  magnetic  circuit 
transformer,  has  a  circuit  of  low  reluctance  —  a  magnetic 
cross-flux  passes  between  the  primary  and  secondary  coils, 
surrounding  one  coil  only,  without  being  interlinked  with 
the  other.     This  magnetic  cross-flux  is  proportional  to  the 
current  flowing  in  the  electric  circuit,  or  rather,  the  ampere- 
turns  or  M.M.F.  increase  with  the  increasing  load  on  the 
transformer,  and  constitute  what   is  called  the   self-induc- 
tance of  the  transformer ;  while  the  flux  surrounding  both 


194  ALTERNATING-CURRENT  PHENOMENA. 

coils  may  be  considered  as  mutual  inductance.  This  cross- 
flux  of  self-induction  does  not  induce  E.M.F.  in  the  second- 
ary circuit,  and  is  thus,  in  general,  objectionable,  by  causing 
a  drop  of  voltage  and  a  decrease  of  output.  It  is  this 
cross-flux,  however,  or  flux  of  self-inductance,  which  is  uti- 
lized in  special  transformers,  to  secure  automatic  regulation, 
for  constant  power,  or  for  constant  current,  and  in  this 
case  is  exaggerated  by  separating  primary  and  secondary 
coils.  In  the  constant  potential  transformer  however,  the 
primary  and  secondary  coils  are  brought  as  near  together  as 
possible,  or  even  interspersed,  to  reduce  the  cross-flux. 

As  will  be  seen  by  the  self-inductance  of  a  circuit,  not 
the  total  flux  produced  by,  and  interlinked  with,  the  circuit 
is  understood,  but  only  that  (usually  small)  part  of  the  flux 
which  surrounds  one  circuit  without  interlinking  with  the 
other  circuit. 

128.  The  alternating  magnetic  flux  of  the  magnetic 
circuit  surrounding  both  electric  circuits  is  produced  by 
the  combined  magnetizing  action  of  the  primary  and  of  the 
secondary  current. 

This  magnetic  flux  is  determined  by  the  E.M.F.  of  the 
transformer,  by  the  number  of  turns,  and  by  the  frequency. 
If 

<£  =  maximum  magnetic  flux, 
N=  frequency, 
n  =  number  of  turns  of  the  coil  ; 

the  E.M.F.  induced  in  this  coil  is 

E=  V2  *•  JVfc  *  10  -8  =  4.44  .Afo*  10  -'volts; 


hence,  if  the  E.M.F.,  frequency,  and  number  of  turns  are 
determined,  the  maximum  magnetic  flux  is 


To   produce  the  magnetism,   $,  of    the  transformer,  a 
M.M.F.  of  5  ampere-turns  is  required,  which  is  determined 


ALTERNATING-CURRENT   TRANSFORMER.  195 

by  the  shape  and  the  magnetic  characteristic  of  the  iron,  in 
the  manner  discussed  in  Chapter  X. 

For  instance,  in  the  closed  magnet  circuit  transformer, 
the  maximum  magnetic  induction  is  ($>  =  &  /S,  where  S 
=  the  cross-section  of  magnetic  circuit. 

129.  To  induce  a  magnetic  density,  ($>,  a  M.M.F.  of  3CTO 
ampere-turns  maximum  is  required,  or,  3COT  /  V2  ampere- 
turns  effective,  per  unit  length  of  the  magnetic  circuit  ; 
hence,  for  the  total  magnetic  circuit,  of  length,  /, 

/3C 
&  =  —  :r-  ampere-turns  ; 


«       *V2 
where  n  =  number  of  turns. 

At  no  load,  or  open  secondary  circuit,  this  M.M.F.,  CF,  is 
furnished  by  the  exciting  current,  T00,  improperly  called  the 
leakage  current,  of  the  transformer  ;  that  is,  that  small 
amount  of  primary  current  which  passes  through  the  trans- 
former at  open  secondary  circuit. 

In  a  transformer  with  open  magnetic  circuit,  such  as 
the  "hedgehog"  transformer,  the  M.M.F.,  &,  is  the  sum 
of  the  M.M.F.  consumed  in  the  iron  and  in  the  air  part  of 
the  magnetic  circuit  (see  Chapter  X.). 

The  energy  of  the  exciting  current  is  the  energy  con- 
sumed by  hysteresis  and  eddy  currents  and  the  small  ohmic 
loss. 

The  exciting  current  is  not  a  sine  wave,  but  is,  at  least 
in  the  closed  magnetic  circuit  transformer,  greatly  distorted 
by  hysteresis,  though  less  so  in  the  open  magnetic  circuit 
transformer.  It  can,  however,  be  represented  by  an  equiv- 
alent sine  wave,  f00,  of  equal  intensity  and  equal  power  with 
the  distorted  wave,  and  a  wattless  higher  harmonic,  mainly 
of  triple  frequency. 

Since  the  higher  harmonic  is  small  compared  with  the 


196  ALTERNATING-CURRENT  PHENOMENA. 

total  exciting  current,  and  the  exciting  current  is  only  a 
small  part  of  the  total  primary  current,  the  higher  harmonic 
.can,  for  most  practical  cases,  be  neglected,  and  the  exciting 
current  represented  by  the  equivalent  sine  wave. 

This  equivalent  sine  wave,  7^,  leads  the  wave  of  mag- 
netism, 3>,  by  an  angle,  a,  the  angle  of  hysteretic  advance  of 
phase,  and  consists  of  two  components,  —  the  hysteretic 
energy  current,  in  quadrature  with  the  magnetic  flux,  and 
therefore  in  phase  with  the  induced  E.M.F.  =  I00  sin  a;  and 
the  magnetizing  current,  in  phase  with  the  magnetic  fluXj 
and  therefore  in  quadrature  with  the  induced  E.M.F.,  and 
so  wattless,  =  I00  cos  a. 

The  exciting  current,  700,  is  determined  from  the  shape 
and  magnetic  characteristic  of  the  iron,  and  number  of 
turns  ;  the  hysteretic  energy  current  is  — 

Power  consumed  in  the  iron 


I00  sin  a 


Induced  E.M.F. 


130.    Graphically,  the  polar  diagram  of  M.M.Fs.  ot   a 
transformer  is  constructed  thus : 


Fig.  94. 


Let,  in  Fig.  94,  O®  =  the  magnetic  flux  in  intensity  and 
phase  (for  convenience,  as  intensities,  the  effective  values 
are  used  throughout),  assuming  its  phase  as  the  vertical; 


ALTERNATING-CURRENT  TRANSFORMER.  197 

that  is,  counting  the  time  from  the  moment  where  the 
rising  magnetism  passes  its  zero  value. 

Then  the  resultant  M.M.F.  is  represented  by  the  vector 
QS,  leading  O<b  by  the  angle  &O®  =  a. 

The  induced  E.M.Fs.  have  the  phase  180°,  that  is,  are 
plotted  towards  the  left,  and  represented  by  the  vectors 
OZT;  and  OE±. 

If,  now,  ft'  =  angle  of  lag  in  the  secondary  circuit,  due 
to  the  total  (internal  and  external)  secondary  reactance,  the 
secondary  current  II ,  and  hence  the  secondary  M.M.F., 
JF1=  «j  /L,  will  lag  behind  £•[  by  an  angle  ft1,  and  have  the 
phase,  180°  +  ft',  represented  by  the  vector  O^1.  Con- 
structing a  parallelogram  of  M.M.Fs.,  with  Off  as  a  diag- 
onal and  Oif1  as  one  side,  the  other  side  or  O'S0  is  the 
primary  M.M.F.,  in  intensity  and  phase,  and  hence,  dividing 
by  the  number  of  primary  turns,  n0,  the  primary  current  is 
/.-*./*.. 

To  complete  the  diagram  of  E  M.Fs. ,  we  have  now,  — 

In  the  primary  circuit  : 

E.M.F.  consumed  by  resistance  is  70r0,  in  phase  with  fot  and 
represented  by  the  vector  OEr0  • 

E.M.F.  consumed  by  reactance  is  IoX0,  90°  ahead  of  I0,  and 
represented  by  the  vector  OEx0 ; 

E.M.F.  consumed  by  induced  E.M.F.  is  E',  equal  and  oppo- 
site to  E'o,  and  represented  by  the  vector  Off. 

Hence,  the  total  primary  impressed  E.M.F.  by  combina- 
tion of  OEr0,  OEx0,  and  OE'  by  means  of  the  parallelo- 
gram of  E.M.Fs.  is, 

E0  =  ~OE0, 

and  the  difference  of  phase  between  the  primary  impressed 
E.M.F.  and  the  primary  current  is 

ft0  =  E0O50. 
In  the  secondary  circuit : 

Counter  E.M.F.  of  resistance  is  1^  in  opposition  with  Iv 
and  represented  by  the  vector  OJS'r^ ; 


198 


AL  TERNA  TING-CURRENT  PHENOMENA, 


90°  behind   7X,   and 


Counter    E.M.F.   of   reactance  is 
represented  by  the  vector  OE^x^ 

Induced  E.M.Fs.,  E(  represented  by  the  vector  OE-[. 

Hence,  the  secondary  terminal  voltage,  by  combination 
of  OEr^  OEx{  and  OE^  by  means  of  the  parallelogram  of 

E.M.Fs.  is  -==• 

A  =  M»II 

and  the  difference  of  phase  between  the  secondary  terminal 
voltage  and  the  secondary  current  is 


As  will  be  seen  in  the  primary  circuit  the  "  components 
of  impressed  E.M.F.  required  to  overcome  the  counter 
E.M.Fs."  were  used  for  convenience,  and  in  the  secondary 
circuit  the  "counter  E.M.Fs." 


Er, 


Fig.  95.    Transformer  Diagram  with  80°  Lag  in  Secondary  Circuit. 

131.  In  the  construction  of  the  transformer  diagram,  it 
is  usually  preferable  not  to  plot  the  secondary  quantities, 
current  and  E.M.F.,  direct,  but  to  reduce  them  to  corre- 
spondence with  the  primary  circuit  by  multiplying  by  the 
ratio  of  turns,  a  =  n0/  nv  for  the  reason  that  frequently 
primary  and  secondary  E.M.Fs.,  etc.,  are  of  such  different 


AL  TERA?A  TING-CURRENT  TRANSFORMER. 


19!) 


magnitude  as  not  to  be  easily  represented  on  the  same 
scale;  or  the  primary  circuit  may  be  reduced  to  the  sec- 
ondary in  the  same  way.  In  either  case,  the  vectors  repre- 
senting the  two  induced  E.M.Fs.  coincide,  or  OE-^  =  OE^. 


Fig.  96.     Transformer  Diagram  with  50°  Lag  in  Secondary  Circuit. 

Figs.  96  to  107  give  the  polar  diagram  of  a  transformer 
having  the  constants  — 


r0  =  .2  ohms, 
x0  =  .33  ohms, 
f!  =  .00167  ohms, 
*!  =  .0025  ohms, 
g0  =  .0100  mhos, 

for  the  conditions  of  secondary  circuit, 


=  .0173  mhos, 
=  100  volts, 
=  60  amperes, 
=10  degrees.   ? 


20°  lead  in  Fig.  99. 
50°  lead     "         100. 
80°  lead     "         101. 


ft'  =  80°  lag  in  Fig.  95. 

50°  lag  "        96. 

20°  lag  "        97. 

O,  or  in  phase,  "         98. 

As  shown  with  a  change  of  /?/  the  other  quantities  E0,  Iv 
I0,  etc.,  change  in  intensity  and  direction.  The  loci  de- 
scribed by  them  are  circles,  and  are  shown  in  Fig.  102, 
with  the  point  corresponding  to  non-inductive  load  marked. 
The  part  of  the  locus  corresponding  to  a  lagging  secondary 


200  ALTERNATING-CURRENT  PHENOMENA. 


Fig.  97.    Transformer  Diagram  with  20°  Lag  in  Secondary  Circuit 


Fig.  98.    Transformer  Diagram  with  Secondary  Current  in  Phase  with  E.M.F. 


Fig.  99.    Transformer  Diagram  with  20°  Lead  in  Secondary  Current. 


ALTERNATING-CURRENT  TRANSFORMER.  201 


(To  EO 

Fig.  100.    Transformer  Diagram  with  50°  Lead  in  Secondary  Circuit. 


Fig.  101.   Transformer  Diagram  with  80°  Lead  in  Secondary  Circuit. 


Fig.   102. 


202 


AL  TERNA  TING-CURRENT  PHENOMENA. 


current  is  shown  in  thick  full  lines,  and  the  part  correspond- 
ing to  leading  current  in  thin  full  lines. 

132.  This  diagram  represents  the  condition  of  constant 
secondary  induced  E.M.F.,  £"/,  that  is,  corresponding  to  a 
constant  maximum  magnetic  flux. 

By  changing  all  the  quantities  proportionally  from  the 
diagram  of  Fig.  102,  the  diagrams  for  the  constant  primary 
impressed  E.M.F.  (Fig.  103),  and  for  constant  secondary 
terminal  voltage  (Fig.  104),  are  derived.  In  these  cases, 
the  locus  gives  curves  of  higher  order. 


Fig.   103. 


Fig.  105  gives  the  locus  of  the  various  quantities  when 
the  load  is  changed  from  full  load,  /j  =  60  amperes  in  a 
non-inductive  secondary  external  circuit  to  no  load  or  open 
circuit. 

a.)  By  increase  of  secondary  resistance  ;  b.}  by  increase 
of  secondary  inductive  reactance ;  c.)  by  increase  of  sec- 
ondary capacity  reactance. 

As  shown  in  a.),  the  locus  of  the  secondary  terminal  vol- 
tage, J5lt  and  thus  of  E0,  etc.,  are  straight  lines;  and  in 
b.)  and  c.},  parts  of  one  and  the  same  circle  a.}  is  shown 


AL  TERNA  TING-CURRENT  TRANSFORMER. 


203 


in  full  lines,  b.}  in  heavy  full  lines,  and  c.}  in  light  full  lines. 
This  diagram  corresponds  to  constant  maximum  magnetic 
flux ;  that  is,  to  constant  secondary  induced  E.M.F.  The 
diagrams  representing  constant  primary  impressed  E.M.F. 
and  constant  secondary  terminal  voltage  can  be  derived 
from  the  above  by  proportionality. 


Fig.   104. 


133.  It  must  be  understood,  however,  that  for  the  pur- 
pose of  making  the  diagrams  plainer,  by  bringing  the  dif- 
ferent values  to  somewhat  nearer  the  same  magnitude,  the 
constants  chosen  for  these  diagrams  represent,  not  the  mag- 
nitudes found  in  actual  transformers,  but  refer  to  greatly 
exaggerated  internal  losses. 

In  practice,  about  the  following  magnitudes  would  be 
found  : 


r0  =  .01  ohms ; 
x0  =  .033  ohms ; 
ri  =  .00008  ohms  j 


#!  =  .00025  ohms  ; 
g0  =  .001  ohms  ; 
b0  =  .00173  ohms ; 


that  is,  about  one-tenth  as  large  as  assumed.  Thus  the 
changes  of  the  values  of  E0,  Elt  etc.,  under  the  different 
conditions  will  be  very  much  smaller. 


204 


ALTERNATING-CURRENT  PHENOMENA. 


Symbolic  Method. 

134.  In  symbolic  representation  by  complex  quantities 
the  transformer  problem  appears  as  follows  : 

The  exciting  current,  700,  of  the  transformer  depends 
upon  the  primary  E.M.F.,  which  dependance  can  be  rep- 
resented by  an  admittance,  the  "  primary  admittance," 
°f  tne  transformer. 


Fig.    105. 

The  resistance  and  reactance  of  the  primary  and  the 
secondary  circuit  are  represented  in  the  impedance  by 

Z0=r0-  jx0,         and         Zl=rl-  j  xl . 

Within  the  limited  range  of  variation  of  the  magnetic 
density  in  a  constant  potential  transformer,  admittance  and 
impedance  can  usually,  and  with  sufficient  .exactness,  be 
considered  as  constant. 

Let 

n0  =  number  of  primary  turns  in  series ; 
#1   =  number  of  secondary  turns  in  series ; 
a     =  —  =  ratio  of  turns ; 

Y0  =  g0  4-  jb0  =  primary  admittance 

Exciting  current  .  ~i       I 

Primary  counter  E.M.F. ' 


.VVWvVl 


rw^ww 


ALTERNATING-CURRENT  TRANSFORMER.  205 

Z0  =  r0  —  j x0  =  primary  impedance  7. — — 

E.M.F.  consumed  in  primary  coil  by  resistance  and  reactance.         ^     -n-f  '"         ' j**/\. 

Primary  current  ~      / 

Z±  =  r±  —jx1=  secondary  impedance 

__  E.M.F.  consumed  in  secondary  coil  by  resistance  and  reactance  . 
Secondary  current 

where  the  reactances,  x0  and  ^ ,  refer  to  the  true  self -induc- 
tance only,  or  to  the  cross-flux  passing  between  primary  and 
secondary  coils  ;  that  is,  interlinked  with  one  coil  only. 
Let  also 

Y    =  g -\-jb-  total    admittance   of   secondary   circuit, 

including  the  internal  impedance  ; 
E0  =  primary  impressed  E.M.F. ; 
E  '  =  E.M.F.  consumed  by  primary  counter  E.M.F. ; 
£i   =  secondary  terminal  voltage  ; 
EI   =  secondary  induced  E.M.F. ; 
I0    =  primary  current,  total ; 
/oo   =  primary  exciting  current ; 
/i     =  secondary  current. 

Since  the  primary  counter  E.M.F.,-£"',  and  the  second- 
ary induced  E.M.F.,  E^,  are  proportional  by  the  ratio  of 

turns,  a, 

E  '  =  —  a  E{.  (1) 

The  secondary  current  is  : 

/i     =  **/,  (2) 

consisting  of  an  energy  component,  gE^,  and  a  reactive 
component,  b  E^. 

To  this  secondary  current  corresponds  the  component  of 
primary  current, 

•7o    =     ~a  a* 

The  primary  exciting  current  is  — 

I«>=YOE>.  (4) 

Hence,  the  total  primary  current  is  : 


206  AL  TERNA  TING-CURRENT  PHENOMENA. 

(6) 


The  E.M.F.  consumed  in  the  secondary  coil  by  the 
internal  impedance  is  Z-J^. 

The  E.M.F.  induced  in  the  secondary  coil  by  the  mag- 
netic flux  is  EI. 

Therefore,  the  secondary  terminal  voltage  is 


or,  substituting  (2),  we  have 

£,  =  £,'  {I-  Z,Y}  (7) 

The  E.M.F.  consumed  in  the  primary  coil  by  the  inter- 
nal impedance  is  Z0  I0. 

The  E.M.F.  consumed  in  the  primary  coil  by  the  counter 
E.M.F.  is  E'. 

Therefore,  the  primary  impressed  E.M.F.  is 

E0  =  E'  +  Z0S0, 
or,  substituting  (6), 


(8) 

\°/ 


136.    We  thus  have, 

primary  E.M.F.,      E0  =  -  aE{  j  1  +  Z0  Y0  +  ^Z  J  ,            (8) 

secondary  E.M.F.,  E^  =  E{  {  1  -  Zl  Y},  (7) 

primary  current,      I0  =  —  — -{Y+a*Y0},  (6) 

secondary  current,  /i  =  YEl't  (2) 

as  functions  of  the  secondary  induced  E.M.F.,  EJ,  as  pa- 
rameter. 


ALTERNATING-CURRENT   TRANSFORMER.  207 

From  the  above  we  derive 

Ratio  of  transformation  of  E.M.Fs.  : 


.       1-Z.K 

Ratio  of  transformations  of  currents  : 


(10) 


From    this    we    get,    at    constant    primary    impressed 
E.M.F., 

E0  =  constant ; 

secondary  induced  E.M.F., 


E.M.F.  induced  per  turn, 
E  1 

n0    -\    \    7  y    \ 

secondary  terminal  voltage, 


primary  current, 

^  4-   Y 
,     .          EA         Y+a*Y0          _  w          ^^    y° 


secondary  current, 

Y 


At  constant  secondary  terminal  voltage, 
-fi1!  =  const. ; 


208  AL  TERNA  TING-CURRENT  PHENOMENA. 

secondary  induced  E.M.F., 

F1    -  £l 

1-^F' 
E.M.F.  induced  per  turn, 


^1-Z.F' 
primary  impressed  E.M.F., 


primary  current, 

/ 

secondary  current, 


136.    Some  interesting  conclusions  can  be  drawn  from 
these  equations. 

The  apparent  impedance  of  the  total  transformer  is 


(14) 


Substituting  now,   —  =  V,  the  total  secondary  admit- 

tance, reduced  to  the  primary  circuit  by  the  ratio  of  turns, 
it  is 


Y0-\-Y'  is  the  total  admittance  of  a  divided  circuit  with 
the  exciting  current,  of  admittance  Y0,  and  the  secondary 


AL  TERN  A  TING-CURRENT  TRANSFORMER. 


209 


current,  of  admittance  Y1  (reduced  to  primary),  as  branches. 
Thus  : 


is  the  impedance  of  this  divided  circuit,  and 


That  is  : 


(17) 


The  alternate-current  transformer,  of  primary  admittance 
Y0  ,  total  secondary  admittance  Y,  and  primary  impedance 
Z0  ,  is  equivalent  to,  and  can  be  replaced  by,  a  divided  circuit 
with  the  branches  of  admittance  Y0  ,  the  exciting  current,  and 
admittance  Y'  =  Y/a2,  the  secondary  current,  fed  over  mains 
of  the  impedance  Z0,  the  internal  primary  impedance. 

This  is  shown  diagrammatically  in  Fig.  106. 


Yog 

z 


Fig.  106. 


137.  Separating  now  the  internal  secondary  impedance 
from  the  external  secondary  impedance,  or  the  impedance  of 
the  consumer  circuit,  it  is 

4 -£.+  *!  (18) 


where  Z  =  external  secondary  impedance, 


(19) 


210  ALTERNATING-CURRENT  PHENOMENA. 

Reduced  to  primary  circuit,  it  is 


=  Z/  +  Z7.  (20) 

That  is  : 

An  alternate-current  transformer,  of  primary  admittance 
Y0,  primary  impedance  Z0,  secondary  impedance  Zv  and 
ratio  of  turns  a,  can,  when  the  secondary  circuit  is  closed  by 
an  impedance  Z  (the  impedance  of  the  receiver  circuit),  be 
replaced,  and  is  equivalent  to  a  circtiit  of  impedance  Z  '  = 
a?Z,  fed  over  mains  of  the  impedance  Z0-\-  Z^,  where  Z^  = 
a2Zlt  shunted  by  a  circuit  of  admittance  Y0,  which  latter 
circuit  branches  off  at  the  points  a  —  b,  between  the  impe- 
dances Z  and  Z-. 


Generator          I,  Transformer    I 


Fig.   107. 

This  is  represented  diagrammatically  in  Fig.  107. 

It  is  obvious  therefore,  that  if  the  transformer  contains 
several  independent  secondary  circuits  they  are  to  be  con- 
sidered as  branched  off  at  the  points  a,  i,  in  diagram 
Fig.  107,  as  shown  in  diagram  Fig.  108. 

It  therefore  follows : 

An  alternate-current  transformer,  of  x  secondary  coils,  of 
the  internal  impedances  Z^,  Z^1,  .  .  .  Z-f,  closed  by  external 
secondary  circuits  of  the  impedances  Z1,  Zn,  .  .  .  Zx,  is  equiv- 
alent to  a  divided  circuit  of  x  +  1  branches,  one  branch  of 


AL  TERN  A  TING-CURRENT  TRANSFORMER. 
Generator  Transformer 


211 


Fig.  108. 

admittance  Y0)  the  exciting  current,  the  other  branches  of  the 
impedances  ZJ  +  Z7,  ZJ1  +  Zn,  .  .  .  2f  +  Zx,  the  latter 
impedances  being  reduced  to  the  primary  circuit  by  the  ratio 
of  turns,  and  the  whole  divided  circuit  being  fed  by  the 
primary  impressed  E.M.F.  £0,  over  -mains  of  the  impedance 
Z0- 

Consequently,  transformation  of  a  circuit  merely  changes 
all  the  quantities  proportionally,  introduces  in  the  mains  the 
impedance  Z0  +  Z^,  and  a  branch  circuit  between  Z0  and 
Z^,  of  admittance  Y0. 

Thus,  double  transformation  will  be  represented  by  dia- 
gram, Fig.  109. 


212  A  L  TERN  A  TING-  CURRENT  PHENOMENA . 

With  this  the  discussion  of  the  alternate-current  trans- 
former ends,  by  becoming  identical  with  that  of  a  divided 
circuit  containing  resistances  and  reactances. 

Such  circuits  have  explicitly  been  discussed  in  Chapter 
VIII.,  and  the  results  derived  there  are  now  directly  appli- 
cable to  the  transformer,  giving  the  variation  and  the  con- 
trol of  secondary  terminal  voltage,  resonance  phenomena,  etc. 

Thus,  for  instance,  if  Z/  =  Z0,  and  the  transformer  con- 
tains an   additional   secondary  coil,  constantly  closed  by  a 
condenser  reactance  of  such  size  that  this  auxiliary  circuit, 
together  with  the  exciting  circuit,  gives  the  reactance  —  x0,  . 
with  a  non-inductive  secondary  circuit  Z^  =  rv  we  get  the  • 
condition  of  transformation  from  constant  primary  potential 
to  constant  secondary  current,  and  inversely,  as  previously 
discussed. 

Non-inductive    Secondary    Circuit. 

138.  In  a  non-inductive  secondary  circuit,  the  external 
secondary  impedance  is, 


or,  reduced  to  primary  circuit, 

Assuming  the   secondary   impedance,   reduced  to  primary 
circuit,  as  equal  to  the  primary  impedance, 

*  is>    Y  '  i     r 


Substituting  these  values  in  Equations  (9),  (10),  and  (13), 
we  have 

Ratio  of  E.M.Fs.  : 


(r0  —  jx0} 
4-  ra—jx0 


ALTERNATING-CURRENT  TRANSFORMER.          213 


+       r0-jx0  f      r0-jx0      Y|   .  .  .  \  . 

R  +  r0  —  jx0       \  R  +  rn  —  /#„ 


or,  expanding,  and  neglecting  terms  of  higher  than  third 
order, 

—  jx0 

^ 


or,  expanded, 

J|=  -  «  1  1  +  2  r°  ^'^  +  (r,  -y^)(.% 

Neglecting  terms  of  tertiary  order  also, 


£t 

Ratio  of  currents  : 

^-  =  -  - 

/I  ^ 

or,  expanded, 

~=-- 

/!  a 

Neglecting  terms  of  tertiary  order  also, 


Total  apparent  primary  admittance  : 


R  +  r0—  jx 
(r0  -jx0}  +  R  (r0- 


=  {R  +  2  (r0  -  y  x0}  -  &  (go  +jb0}  -2  R  (r0  -  Jx0) 


214  ALTERNATING-CURRENT  PHENOMENA. 

or, 

b0}-  2  (r0  -Jx0}( 


Neglecting  terms  of  tertiary  order  also  : 
Zt=R 

Angle  of  lag  in  primary  circuit : 

tan  S>0  =  ^ ,  hence, 
rt 

2^+Rb0  +  2r0b0-2Xogo-2 
tan  S>0  =  a 


Neglecting  terms  of  tertiary  order  also  : 
'R 


139.  If,  now,  we  represent  the  external  resistance  of 
the  secondary  circuit  at  full  load  (reduced  to  the  primary 
circuit)  by  R0,  and  denote, 


2  r0   _        _          .  Internal  resistance  of  transformer         _  percentage 

R0    ~  External  resistance  of  secondary  circuit  ~  na^  resistance, 

2  X0   _        __  ratjQ       Internal  reactance  of  transformer          _  percentage 
J£  '       External  resistance  of  secondary  circuit  nal  reactance 

X*.-  h  -  ratio  -  percentage  hysteresis, 

,,  ,  ,  .      Magnetizing  current        percentage  magnetizing  cur- 

•KO  °o=  g  =      -10  Totalsecondarycurrent  =  rent^ 

and  if  d  represents  the  load  of  the  transformer,  as  fraction 
of  full  load,  we  have 


ALTERNATING-CURRENT  TRANSFORMER.  215 


and, 


**.-«. 

a 

Substituting  these  values  we  get,  as  the  equations  of  the 
transformer  on  non-inductive  load, 
Ratio  of  E.M.Fs.  : 


or,  eliminating  imaginary  quantities, 


H"-"^) 


Ratio  of  currents  : 

+  (h  +> 

d 


2  f 

.  ^ 

or,  eliminating  imaginary  quantities, 


1  f 

a  \ 


i  i  h  i 


216  ALTERNATING-CURRENT  PHENOMENA. 

Total  apparent  primary  impedance  : 
Z,  = 


or,  eliminating  imaginary  quantities, 


Angle  of  lag  in  primary  circuit : 


That  is, 

An  alternate-current  transformer,  feeding  into  a  non-induc- 
tive secondary  circuit,  is  represented  by  the  constants  : 

R0  =  secondary  external  resistance  at  full  load ; 

p    =  percentage  resistance  ; 

q    =  percentage  reactance  ; 

h     =  percentage  hysteresis  ; 

g    =  percentage  magnetizing  current ; 

d    =  secondary  percentage  load. 

All  these  qualities  being  considered  as  reduced  to  the  primary 
circuit  by  the  square  of  the  ratio  of  turns,  a2. 


ALTERNATING-CURRENT  TRANSFORMER. 


217 


140.    As   an  instance,  a  transformer  of  the  following 
constants  may  be  given  : 


e0    =1,000; 
a     =        10  ; 


£0=     120; 

p    =  .02  • 


q  =  .06  ; 
h  =  .02  ; 
g  =  .04. 


Substituting  these  values,  gives  : 
100 


= 

" 


V(i.oou  +  .02  </)2  +  (.0002  +  .06  <ty 


*-^-£- 


.1  ii  V/Y  1.0014  +  —  Y  +  (  —  - 
\\  d  J        \  d 


.  0002    . 


-  -.0004- 


tan  w, 


^- 


1.9972  +  . 


Fig.   110.     Load  Diagram  of  Transformer. 


218  ALTERNATING-CURKENT  PHENOMENA. 

In  diagram  Fig.  110  are  shown,  for  the  values  from 
d  =  0  to  d=  1.5,  with  the  secondary  current  ix  as  abscis- 
sae, the  values : 

secondary  terminal  voltage,  in  volts, 

secondary  drop  of  voltage,  in  per  cent, 

primary  current,  in  amps, 

excess    of    primary    current    over    proportionality    with 

secondary,  in  per  cent, 
primary  angle  of  lag. 

The  power-factor  of  the  transformer,  cos  w0,  is  .45  at 
open  secondary  circuit,  and  is  above  .99  from  25  amperes, 
upwards,  with  a  maximum  of  .995  at  full  load. 


ALTERNATING-CURRENT   TRANSFORMER.  219 


CHAPTER    XV. 

THE  GENERAL  ALTERNATING-CURRENT  TRANSFORMER  OR 
FREQUENCY  CONVERTER. 

141.  The  simplest  alternating-current  apparatus  is  the 
alternating-current  transformer.  It  consists  of  a  magnetic- 
circuit,  interlinked  with  two  electric  circuits  or  sets  of 
electric  circuits.  The  one,  the  primary  circuit,  is  excited 
by  an  impressed  E.M.F.,  while  in  the  other,  the  secondary 
circuit,  an  E.M.F.  is  induced.  Thus,  in  the  primary  circuit, 
power  is  consumed,  in  the  secondary  circuit  a  correspond- 
ing amount  of  power  produced  ;  or  in  other  words,  power 
is  transferred  through  space,  from  primary  to  secondary 
circuit.  This  transfer  of  power  finds  its  mechanical  equiv- 
alent in  a  repulsive  thrust  acting  between  primary  and 
secondary.  Thus,  if  the  secondary  coil  is  not  held  rigidly 
as  in  the  stationary  transformer,  it  will  be  repelled  and 
move  away  from  the  primary.  This  mechanical  effect  is 
made  use  of  in  the  induction  motor,  which  represents  a 
transformer  whose  secondary  is  mounted  movably  with  re- 
gard to  the  primary  in  such  a  way  that,  while  set  in  rota- 
tion, it  still  remains  in  the  primary  field  of  force.  The 
condition  that  the  secondary  circuit,  while  revolving  with 
regard  to  the  primary,  does  not  leave  the  primary  field  of 
magnetic  force,  requires  that  this  field  is  not  undirectional, 
but  that  an  active  field  exists  in  every  direction.  One  way 
of  producing  such  a  magnetic  field  is  by  exciting  different 
primary  circuits  angularly  displaced  in  space  with  each 
other  by  currents  of  different  phase.  Another  way  is  to 
excite  the  primary  field  in  one  direction  only,  and  get  the 
cross  magnetization,  or  the  angularly  displaced  magnetic 
field,  by  the  reaction  of  the  secondary  current. 


220  ALTERNATING-CURRENT  PHENOMENA. 

We  see,  consequently,  that  the  stationary  transformer 
and  the  induction  motor  are  merely  different  applications 
of  the  same  apparatus,  comprising  a  magnetic  circuit  in- 
terlinked with  two  electric  circuits.  Such  an  apparatus 
can  properly  be  called  a  "general  alternating- current  trans- 
former" The  equations  of  the  stationary  transformer  and 
those  of  the  induction  motor  are  merely  specializations  of 
the  general  alternating-current  transformer  equations. 

Quantitatively  the  main  differences  between  induction 
motor  and  stationary  transformer  are  those  produced  by 
the  air-gap  between  primary  and  secondary,  which  is  re- 
quired to  give  the  secondary  mechanical  movability.  This 
air-gap  greatly  increases  the  magnetizing  current  over  that 
in  the  closed  magnetic  circuit  transformer,  and  requires 
an  ironclad  construction  of  primary  and  secondary  to  keep 
the  magnetizing  current  within  reasonable  limits.  An  iron- 
clad construction  again  greatly  increases  the  self-induction 
of  primary  and  secondary  circuit.  Thus  the  induction 
motor  is  a  transformer  of  large  magnetizing  current  and 
large  self-induction;  that  is,  comparatively  large  primary 
exciting  susceptance  and  large  reactance. 

The  general  alternating-current  transformer  transforms 
between  electrical  and  mechanical  power,  and  changes  not 
only  E.M.Fs.  and  currents,  but  frequencies  also,  and  may 
therefore  be  called  a  "frequency  converter."  Obviously, 
it  also  may  change  the  number  of  phases. 

142.  Besides  the  magnetic  flux  interlinked  with  both 
primary  and  secondary  electric  circuit,  a  magnetic  cross- 
flux  passes  in  the  transformer  between  primary  and  second- 
ary, surrounding  one  coil  only,  without  being  interlinked 
with  the  other.  This  magnetic  cross-flux  is  proportional  to 
the  current  flowing  in  the  electric  circuit,  and  constitutes 
what  is  called  the  self-induction  of  the  transformer.  As 
seen,  as  self-induction  of  a  transformer  circuit,  not  the  total 
flux  produced  by  and  interlinked  with  this  circuit  is  under- 
stood, but  only  that  —  usually  small  —  part  of  the  flux 


AL  TERN  A  TING-CURRENT  TRA  NSFORMER.   221 

which  surrounds  the  one  circuit  without  interlinking  with 
the  other,  and  is  thus  produced  by  the  M.M.F.  of  one 
circuit  only. 

143.  The  mutual  magnetic  flux  of  the  transformer  is 
produced  by  the  resultant  M.M.F.  of  both  electric  circuits. 
It  is  determined  by  the  counter  E.M.F.,  the  number  of 
turns,  and  the  frequency  of  the  electric  circuit,  by  the. 
equation: 


Where  E  =  effective  E.M.F. 

JV=  frequency. 
n    =  number  of  turns. 
<£  ==  maximum  magnetic  flux. 

The  M.M.F.  producing  this  flux,  or  the  resultant  M.M.F. 
of  primary  and  secondary  circuit,  is  determined  by  shape 
and  magnetic  characteristic  of  the  material  composing  the 
magnetic  circuit,  and  by  the  magnetic  induction.  At  open 
secondary  circuit,  this  M.M.F.  is  the  M.M.F.  of  the  primary 
current,  which  in  this  case  is  called  the  exciting  current, 
and  consists  of  an  energy  component,  the  magnetic  energy 
current,  and  a  reactive  component,  the  magnetizing  current. 

144.  In    the   general    alternating-current    transformer, 
where  the  secondary  is  movable  with  regard  to  the  primary, 
the  rate  of  cutting  of  the  secondary  electric  circuit  with  the 
mutual  magnetic  flux  is  different  from  that  of  the  primary. 
Thus,  the  frequencies  of  both  circuits  are  different,  and  the 
induced   E.M.Fs.   are   not   proportional  to  the  number  of 
turns  as  in  the  stationary  transformer,  but  to  the  product 
of  number  of  turns  into  frequency. 

145.  Let,  in  a  general  alternating-current  transformer  : 

*  =  ratio  iS^  frequency,  or  «  slip  »  ; 
thus,  if 

N  '=  primary  frequency,  or  frequency  of  impressed  E.M.F., 
s  JV  =  secondary  frequency  ; 


222  ALTERNATING-CURRENT  PHENOMENA. 

and  the  E.M.F.  induced  per  secondary  turn  by  the  mutual 
flux  has  to  the  E.M.F.  induced  per  primary  turn  the  ratio  s, 

s  =  0  represents  synchronous  motion  of  the  secondary ; 

s  <  0  represents  motion  above  synchronism — driven  by  external 

mechanical  power,  as  will  be  seen  ; 
s  =  1  represents  standstill ; 
s  >  1  represents  backward  motion  of  the  secondary 

that  is,  motion  against  the  mechanical  force  acting  between 
primary  and  secondary  (thus  representing  driving  by  ex- 
ternal mechanical  power). 
Let 

«0   =  number  of  primary  turns  in  series  per  circuit ; 

/?!   =  number  of  secondary  turns  in  series  per  circuit ; 

a     =  —  =  ratio  of  turns  ; 
«i 

Y0  =£"0  H~./A)  =  primary  exciting  admittance  per  circuit; 

where 

gQ    =  effective  conductance  ; 

b0    =  susceptance  ; 

Z0  =  r0  —jx0  =  internal  primary  self-inductive  impedance 

per  circuit, 
where 

r0    =  effective  resistance  of  primary  circuit ; 

jr0   =  reactance  of  primary  circuit ; 

Zu  =  TI  —  jxv  =  internal  secondary  self -inductive  impedance 

per  circuit  at  standstill,  or  for  s  =  1, 
where 

rj    =  effective  resistance  of  secondary  coil ; 
Xl   —  reactance  of  secondary  coil  at  standstill,  or  full  fre- 
quency, s  =  1. 

Since  the  reactance  is  proportional  to  the  frequency,  at 
the  slip  s,  or  the  secondary  frequency  s  N,  the  secondary 

impedance  is : 

Zl  =  r1-jsxl. 

Let  the  secondary  circuit  be  closed  by  an  external  re- 
sistance r,  and  an  external  reactance,  and  denote  the  latter 


ALTERNATING-CURRENT  TRANSFORMER,          223 

by  x  at  frequency  N,  then  at  frequency  s  N,  or  slip  s,  it 
will  be  =  s  x,  and  thus  : 

Z  =  r  —  jsx  =  external  secondary  impedance.* 
Let 

£0  =  primary  impressed  E.M.F.  per  circuit, 
E  '  =  E.M.F.  consumed  by  primary  counter  E.M.F., 
£1   =  secondary  terminal  E.M.F., 
EI   =  secondary  induced  E.M.F., 
e      =  E.M.F.  induced  per  turn  by  the  mutual  magnetic  flux, 

at  full  frequency  JY, 
IQ    =  primary  current, 
^0  =  primary  exciting  current, 
7i    =  secondary  current. 

It  is  then  : 

Secondary  induced  E.M.F. 

EI  =  sn^e. 

Total  secondary  impedance 

Z,  +  Z=  (r,  +  r) 
hence,  secondary  current 


Secondary  terminal  voltage 


*  This  applies  to  the  case  where  the  secondary  contains  inductive  reac- 
tance only  ;  or,  rather,  that  kind  of  reactance  which  is  proportional  to  the  fre- 
quency. In  a  condenser  the  reactance  is  inversely  proportional  to  the  frequency, 
in  a  synchronous  motor  under  circumstances  independent  of  the  frequency. 
Thus,  in  general,  we  have  to  set,  x  =  x'  +  x"  -\  x"\  where  x'  is  that  part  of 
the  reactance  which  is  proportional  to  the  frequency,  x"  that  part  of  the  reac- 
tance independent  of  the  frequency,  and  x'"  that  part  of  the  reactance  which 
is  inversely  proportional  t6  the  frequency  ;  and  have  thus,  at  slip  s,  or  frequency 
sN,  the  external  secondary  reactance  sx'  +  x"  -f-  —  —  . 


224  AL  TERNA  TING-CURRENT  PHENOMENA, 

E.M.F.  consumed  by  primary  counter  E.M.F. 

£'=  -«<>'; 

hence,  primary  exciting  current : 

700  =  E  '  YQ  =  —  «0  e  (g0  +  /£<))• 

Component  of  primary  current  corresponding  to  second- 
ary current  7X : 


hence,  total  primary  current, 

// 
1 


Primary  impressed  E.M.F., 


We  get  thus,  as  the 
Equations  of  the  General  Alternating-Current  Transformer: 

Of  ratio  of  turns,  a  ;  and  ratio  of  frequencies,  s  ;  with  the 
E.M.F.  induced  per  turn  at  full  frequency,  e,  as  parameter, 
the  values  : 

Primary  impressed  E.M.F., 


Secondary  terminal  voltage, 


Primary  current, 

\  1 


ALTERNATING-CURRENT  TRANSFORMER.          225 


Secondary  current, 

II    =7— -7- 


Therefrom,  we  get  : 
Ratio  of  currents, 


Ratio  of  E.M.Fs., 


Total  apparent  primary  impedance, 


,  ,    .    x"    .  x'" 

where  x—x-\  ---  \-  — 

s         s2 

in  the  general  secondary  circuit  as  discussed  in  foot-note, 
page  221. 

Substituting  in  these  equations  : 

*-l, 

gives  the 

General  Equations  of  the  Stationary  Alternating-Current 
Transformer  : 


z*+z\         z,  +  z 

'*      =    -»•<     \        .,;,*  +IU- 

»*  (Zj  +  Z) 


ALTERNA TING-CURRENT  PHENOMENA. 


r  nte 

yi    = 


Z,  +  Z 
/!  a 

P                   f1  +    *f7\^  +  Z'Y* 
^o_=  _  a  }          a  (Z-j  +  2} 

&  I-      Z* 

(  Z,  +  Z 


1+       2//°  x+^oKo] 

a2  (Zj  +  Z)  _  I 

l  +  ^Fo^  +  Z)     J 


Substituting  in  the  equations  of  the  general  alternating- 
current  transformer, 

Z  =  0, 
gives  the 

General  Eqtiations  of  tJie  Induction  Motor: 


a'r^-jsx^ 
^  =  0. 

1       i  ^o  +y^o 


70  =  _  s  «0  f  ]  -T,          .      . 

1  «•(>-!— y**o 


r       j«,^ 

A  = — 


—5  "^ : — ~  +  (ro  — y^o)(^b  +/ 

«2^i  —  JSXi 


Returning  now  to  the  general  alternating-current  trans^ 
former,  we  have,  by  substituting 

(ri  +  r?  +  ^2  (*i  +  *)2  =  **f, 
and  separating  the  real  and  imaginary  quantities, 

-±-  (r0  (r,  +  r)+sx9(Xl  +  x)) 
22 


ALTERNATING-CURRENT  TRANSFORMER,          227 


Neglecting  the  exciting  current,  or  rather  considering 
it  as  a  separate  and  independent  shunt  circuit  outside  of 
the  transformer,  as  can  approximately  be  done,  and  assum- 
ing the  primary  impedance  reduced  to  the  secondary  circuit 
as  equal  to  the  secondary  impedance, 


Substituting  this  in  the  equations  of  the  general  trans- 
former, we  get, 

£,=  -  «0  e\  I  +     -  fr  fa  +  r) 


146.    The  true  power  is,  in  symbolic  representation  (see 
Chapter  XII.)  : 


228  ALTERNATING-CURRENT  PHENOMENA. 

denoting, 

safe* 

-7F  =  W 

gives : 

Secondary  output  of  the  transformer 


Internal  loss  in  secondary  circuit, 

m        -2  t  s  n\  ^\2 

-Pi    =  'i2  n  =  (  —  —  } 

V  **  / 

Total  secondary  power, 


** 

Internal  loss  in  primary  circuit, 

r»i        -9  -9o 

^o  =  V'o  =  4  rt<r 

Total  electrical  output,  plus  loss, 

2 


Total  electrical  input  of  primary, 


Hence,  mechanical  output  of  transformer, 

P=P»-P*  =  w(l-s)(r 
E.atio, 


mechanical 


output      _  P  1  —  S  _  speed 


total  secondary  power         P   -\-  P  l 

147.    Thus, 

In  a  general  alternating  transformer  of  ratio  of  turns,  a, 
and  ratio  of  frequencies,  s,  neglecting  exciting  current,  it  is  : 

Electrical  input  in  primary, 
P 


ALTERNATING-CURRENT  TRANSFORMER.          229 

Mechanical  output, 

P  -   jg-j)«iV(r+rO 
' 


Electrical  output  of  secondary, 


Losses  in  transformer, 


Of  these  quantities,  P1  and  Pl  are  always  positive ;  PQ 
and  P  can  be  positive  or  negative,  according  to  the  value 
of  s.  Thus  the  apparatus  can  either  produce  mechanical 
power,  acting  as  a  motor,  or  consume  mechanical  power; 
and  it  can  either  consume  electrical  power  or  produce 
electrical  power,  as  a  generator. 

148.   At 

s  =  0,  synchronism,  PQ  =  0,  P  =  0,  Pl  =  0. 
At     0  <  s  <  1,  between  synchronism  and  standstill. 

Pl ,  P  and  PQ  are  positive ;  that  is,  the  apparatus  con- 
sumes electrical  power  PQ  in  the  primary,  and  produces 
mechanical  power  P  and  electrical  power  Pl  -j-  P^  in  the 
secondary,  which  is  partly,  P-^,  consumed  by  the  internal 
secondary  resistance,  partly,  Pl ,  available  at  the  secondary 
terminals. 

In  this  case  it  is : 

•Pi  +  ^i1  _      J 
P        ~l-s> 

that  is,  of  the  electrical  power  consumed  in  the  primary 
circuit,  P0,  a  part  P^  is  consumed  by  the  internal  pri- 
mary resistance,  the  remainder  transmitted  to  the  secon- 
dary, and  divides  between  electrical  power,  P1  +  P^1,  and 
mechanical  power,  P,  in  the  proportion  of  the  slip,  or  drop 
below  synchronism,  s,  to  the  speed :  1  —  s. 


230  ALTERNATING-CURRENT  PHENOMENA. 

In  this  range,  the  apparatus  is  a  motor. 
At  s  >  1 ;  or,  backwards  driving, 

P  <  0,  or  negative  ;  that  is,  the  apparatus  requires  mechanical 
power  for  driving. 

It  is  then  :  P0  -  A1  -  A1  <  PI  ; 

that  is  :  the  secondary  electrical  power  is  produced  partly 
by  the  primary  electrical  power,  partly  by  the  mechanical 
power,  and  the  apparatus  acts  simultaneously  as  trans- 
former and  as  alternating-current  generator,  with  the  sec- 
ondary as  armature. 

The  ratio  of  mechanical  input  to  electrical  input  is  the 
ratio  of  speed  to  synchronism. 

In  this  case,  the  secondary  frequency  is  higher  than  the 
primary. 

At  s  <  0,  beyond  synchronism, 

P  <  0 ;  that  is,  the  apparatus  has  to  be  driven  by  mechanical 

power. 
/o<0;    that  is,  the  primary  circuit  produces  electrical  power 

from  the  mechanical  input. 

At  r+r!  +  srj.  =  0,    or,    s  <  —  ^±^J ; 

rt 

the  electrical  power  produced  in  the  primary  becomes  less 
than  required  to  cover  the  losses  of  power,  and  />0  becomes 
positive  again. 
We  have  thus : 

K-£±fl 
r\ 

consumes  mechanical  and  primary  electric  power ;  produces 
secondary  electric  power. 

-  r-±^  <  s  <  0 
?i 

consumes  mechanical,  and  produces  electrical  power  in 
primary  and  in  secondary  circuit. 


ALTERNATING-CURRENT  TRANSFORMER.  231 

consumes  primary  electric  power,  and  produces  mechanical 
and  secondary  electrical  power. 


consumes   mechanical  and  primary  electrical  power ;  pro- 
duces secondary  electrical  power. 


T 


GENERAL  ALTERNATE  CURRENT  TRANSFORMER 


A 


648 

Fig 


H 


149.    As  an  instance,  in  Fig.  Ill  are  plotted,  with  the 
slip  s  as  abscissae,  the  values  of : 

Secondary  electrical  output  as  Curve      I.  ; 
Total  internal  loss  as  Curve    II.  ; 

Mechanical  output  as  Curve  III.  ; 

Primary  electrical  input        as  Curve  IV. ; 

for  the  values  : 

n,e  =  100.0 ;  r     =         A ; 

r»  —        4.  i  x    =        .3; 


232  ALTERNATING-CURRENT  PHENOMENA. 

hence,  p    =  16,000  ^2. 

pl,    Pi  _  8,000  j«. 

0  """     l -i    , — j" ? 

„    _  4,000  s  +  (5  + J)  . 

~  1       I         2 ' 

p    =  20,000  s  (1  -  j) 

150.  Since  the  most  common  practical  application  of 
the  general  alternating  current  transformer  is  that  of  fre- 
quency converter,  that  is  to  change  from  one  frequency  to 
another,  either  with  or  without  change  of  the  number  of 
phases,  the  following  characteristic  curves  of  this  apparatus 
are  of  great  interest. 

1.  The  regulation  curve ;  that  is,  the  change  of  second- 
ary terminal  voltage  as  function  of  the  load  at  constant  im- 
pressed primary  voltage. 

2.  The  compounding  curve ;  that  is,  the  change  of  pri- 
mary impressed  voltage  required  to  maintain  constant  sec- 
ondary terminal  voltage. 

In  this  case  the  impressed  frequency  and  the  speed  are 
constant,  and  consequently  the  secondary  frequency.  Gen- 
erally the  frequency  converter  is  used  to  change  from  a  low 
frequency,  as  25  cycles,  to  a  higher  frequency,  as  62.5 
cycles,  and  is  then  driven  backward,  that  is,  against  its 
torque,  by  mechanical  power.  Mostly  a  synchronous  motor 
is  employed,  connected  to  the  primary  mains,  which  by 
over-excitation  compensates  also  for  the  lagging  current  of 
the  frequency  converter. 

Let, 

Y0  =  g0  +j&0  =  primary  exciting  admittance  per  circuit 
of  the  frequency  converter. 

Z^  =  rt  —jx^—  internal  self  inductive  impedance  per 
secondary  circuit,  at  the  secondary  frequency. 


ALTERNATING-CURRENT  TRANSFORMER.  233 

Z^  =  r0  —  jx^  =  internal  self  inductive  impedance  per 
primary  circuit  at  the  primary  frequency. 

a  =  ratio  of  secondary  to  primary  turns  per  circuit. 

b  =  ratio  of  number  of  secondary  to  number  of  primary 
circuits. 

c  =  ratio  of  secondary  to  primary  frequencies. 

Let, 

e  =  induced  E.M.F.  per  secondary  circuit  at  secondary 
frequency. 

Z  =  r  —  jx  =  external  impedance  per  secondary  circuit 
at  secondary  frequency,  that  is  load  on  secondary  system, 
where  x  —  0  for  noninductive  lead. 

We  then  have, 

total  secondary  impedance, 

Z  +  Z1  =  (r-^rl)-j(x  +  x1) 
secondary  current, 


where, 

r  +  r.  x  +  Xl 


(r  +  0>2  +  (*  +  ^)2  (r  +^i)2  +  (*  + 

secondary  terminal  voltage, 

Ei  =  IiZ  =  e  ^4-T 


—  e(r  —jx)  (at 
where, 


primary  induced  E.M.F.  per  circuit, 


primary  load  current  per  circuit, 

71  =  abli  =  abe  (a{ 
primary  exciting  current  per  circuit, 


234  ALTERNATING-CURRENT  PHENOMENA. 

thus,  total  primary  current, 

70  =  71  +  /oo 

=  e  (fi 


where, 

<.  =  •**+£  <•.=«**+! 

primary  terminal  voltage  : 


where, 

d  -—       re        x  d  -re  -x 

ac 

or  absolute, 

e0  =  e  vX2  +  42 

.  =          e°  - 

V^2  +  4« 

substituting   this    value  of   e   in  the  preceding   equations, 
gives,  as  function  of  the  primary  impressed  E.M.F.,  e0: 
secondary  current, 


7  =  >  absolu        7  =    vi 

V4»  +  42  v  ^i2  + 

secondary  terminal  voltage, 


primary  current, 

,  _ 


primary  impressed  E.M.F. 

^     _  ^0   (4 

"  V4 
secondary  output, 


gl^  + 


AL  TERNA  TING-CURRENT  TRANSFORMER. 


235 


primary  electrical  input, 


i  + 


Lr°:oj        </•  +  </.* 

primary  apparent  input,  voltamperes, 
<2o  =  4/o 

Substituting  thus  different  values  for  the  secondary  in- 
ternal impedance  Z  gives  the  regulation  curve  of  the  fre- 
quency converter. 


REGULATION  CURVES 

VOLTS  CONSTANT!  25  CYCLES 


DARY   62.5    CYCLE      QUARTER-PHASE 


TDARY 
20 


CURRENT    PER    PHASE,  AMP. 

3) 


Fig.  112, 

Such  a  curve,  taken  from  tests  of  a  20"0  KW  frequency 
converter  changing  from  6300  volts  25  cycles  three-phase, 
to  2500  volts  62.5  cycles  quarter-phase,  is  given  in  Fig. 
112. 


236  AL  TERN  A  TING-CURRENT  PHENOMENA. 

From  the  secondary  terminal  voltage, 


it  follows,  absolute, 


PRIlt 

ARY 

VOUT8 

AMP. 



=^ 

• 

/ 

J500 
6000 

J3- 
12 

/ 

11 

/ 

/ 

in 

/ 

/ 

9 

x 

8 

x' 

X 

7 

X 

COND/ 

BY,    2 

( 

00   VO 

OMPC 

LT8C( 

SUND 
NSTAt 

NGC 
T      82 

°.RcV5 

LE3   Q 

ARTE 

F 

6 



^ 

^^ 

PRIM 

kRY,    ! 

5   CYC 

E?  Tl 

REE-P 

HASE 

5 

4 

o 

1 

TDAR 
2 

r  CUR 
} 

1ENT 
| 

PER    P 
0 

HASE, 
4 

AMP. 

t 

.', 

I 

| 

) 

Fig.  113. 


Substituting  these  values  in  tne  above  equation  gives 
the  quantities  as  functions  of  the  secondary  terminal  vol- 
tage, that  is  at  constant  el,  or  the  compounding  curve. 

The  compounding  curve  of  the  frequency  converter 
above  mentioned  is  given  in  Fig.  113. 


INDUCTION  MOTOR.  237 


CHAPTER    XVI. 

INDUCTION  MOTOR. 

151.  A  specialization  of  the  general  alternating-current 
transformer  is  the  induction  motor.  It  differs  from  the 
stationary  alternating-current  transformer,  which  is  also  a 
specialization  of  the  general  transformer,  in  so  far  as  in  the 
stationary  transformer  only  the  transfer  of  electrical  energy 
from  primary  to  secondary  is  used,  but  not  the  mechanical 
force  acting  between  the  two,  and  therefore  primary  and 
secondary  coils  are  held  rigidly  in  position  with  regard  to 
each  other.  In  the  induction  motor,  only  the  mechanical 
force  between  primary  and  secondary  is  used,  but  not  the 
transfer  of  electrical  energy,  and  thus  the  secondary  circuits 
closed  upon  themselves.  Transformer  and  induction  motor 
thus  are  the  two  limiting  cases  of  the  general  alternating- 
current  transformer.  Hence  the  induction  motor  consists 
of  a  magnetic  circuit  interlinked  with  two  electric  circuits  or 
sets  of  circuits,  the  primary  and  the  secondary  circuit,  which 
are  movable  with  regard  to  each  other.  In  general  a  num- 
ber of  primary  and  a  number  of  secondary  circuits  are  used, 
angularly  displaced  around  the  periphery  of  the  motor,  and 
containing  E.M.Fs.  displaced  in  phase  by  the  same  angle. 
This  multi-circuit  arrangement  has  the  object  always  to 
retain  secondary  circuits  in  inductive  relation  to  primary 
circuits  and  vice  versa,  in  spite  of  their  relative  motion. 

The  result  of  the  relative  motion  between  primary  and 
secondary  is,  that  the  E.M.Fs.  induced  in  the  secondary  or 
the  motor  armature  are  not  of  the  same  frequency  as  the 
E.M.Fs.  impressed  upon  the  primary,  but  of  a  frequency 
which  is  the  difference  between  the  impressed  frequency 


238  ALTERNATING-CURRENT  PHENOMENA. 

and  the  frequency  of  rotation,  or  equal  to  the  "slip,"  that  is, 
the  difference  between  synchronism  and  speed  (in  cycles). 
Hence,  if 

N  =  frequency  of  main  or  primary  E.M.F., 

and  s  =  percentage  slip ; 

sJV  =  frequency  of  armature  or  secondary  E.M.F., 

and  (1  —  s)  N=  frequency  of  rotation  of  armature. 

In  its  reaction  upon  the  primary  circuit,  however,  the 
armature  current  is  of  the  same  frequency  as  the  primary 
current,  since  it  is  carried  around  mechanically,  with  a  fre- 
quency equal  to  the  difference  between  its  own  frequency 
and  that  of  the  primary.  Or  rather,  since  the  reaction  of 
the  secondary  on  the  primary  must  be  of  primary  frequency 
— whatever  the  speed  of  rotation — the  secondary  frequency 
is  always  such  as  to  give  at  the  existing  speed  of  rotation  a 
reaction  of  primary  frequency. 

152.  Let  the  primary  system  consist  of  /0  equal  circuits, 
displaced  angulary  in  space  by  1  //0  of  a  period,  that  is, 
1  //„  of  the  width  of  two  poles,  and  excited  by  /»0  E.M.Fs. 
displaced  in  phase  by  1  //0  of  a  period ;  that  is,  in  other 
words,  let  the  field  circuits  consist  of  a  symmetrical  /0-phase 
system.  Analogously,  let  the  armature  or  secondary  circuits 
consist  of  a  symmetrical  /rphase  system. 

Let 

n0  =  number  of  primary  turns  per  circuit  or  phase ; 
«a  =  number  of  secondary  turns  per  circuit  or  phase ; 

a  =  -^  =  ratio  of  total  primary  turns  to  total  secondary  turns 
n\P\ 
or  ratio  of  transformation. 

Since  the  number  of  secondary  circuits  and  number  of 
turns  of  the  secondary  circuits,  in  the  induction  motor  —  as 
in  the  stationary  transformer  —  is  entirely  unessential,  it  is 
preferable  to  reduce  all  secondary  quantities  to  the  primary 
system,  by  the  ratio  of  transformation,  a ;  thus 


INDUCTION  MOTOR.  239 

if  E{  =  secondary  E.M.F.  per  circuit,  El  =  aE{ 

=  secondary  E.M.F.  per  circuit  reduced  to  primary  system; 

if  //    =  secondary  current  per  circuit,  fl=  — 

=  secondary  current  per  circuit  reduced  to  primary  system  ; 
if  r^    =  secondary  resistance  per  circuit,  rt   =  a2  r{ 

=  secondary  resistance  per  circuit  reduced  to  primary  system  ; 
if  x±  =  secondary  reactance  per  circuit,  xt  =  a2  x\ 

=  secondary  reactance  per  circuit  reduced  to  primary  system  ; 
if  £/  =  secondary  impedance  per  circuit,  z1  =  azz\ 

=  secondary  impedance  per  circuit  reduced  to  primary  system  ; 

that  is,  the  number  of  secondary  circuits  and  of  turns  per 
secondary  circuit  is  assumed  the  same  as  in  the  primary 
system. 

In  the  following  discussion,  as  secondary  quantities,  the 
values  reduced  to  the  primary  system  shall  be  exclusively 
used,  so  that,  to  derive  the  true  secondary  values,  these 
quantities  have  to  be  reduced  backwards  again  by  the  factor 

a  =  ?*£-. 
«iA 
153.    Let 

$  =  total  maximum  flux  of  the  magnetic  field  per  motor  pole, 
We  then  have 

E—  V2  77-72  TV^  10  ~8  =  effective  E.M.F.  induced  by  the  mag- 
netic field  per  primary  circuit. 

Counting  the  time  from  the  moment  where  the  rising 
magnetic  flux  of  mutual  induction  &  (flux  interlinked  with 
both  electric  circuits,  primary  and  secondary)  passes  through 
zero,  in  complex  quantities,  the  magnetic  flux  is  denoted  by 


and  the  primary  induced  E.M.F., 


240  ALTERNATING-CURRENT  PHENOMENA. 

where 

e=  V2irrt7V<I>10-8  maybe  considered  as  the  "Active  E.M.F. 
of  the  motor,"  or  "  Counter  E.M.F." 

Since  the  secondary  frequency  is  s  N,  the  secondary  in- 
duced E.M.F.  (reduced  to  primary  system)  is  El  =  —  se. 
Let 

I0  =  exciting  current,  or  current  passing  through  the  motor,  per 
primary  circuit,  when  doing  no  work  (at  synchronism), 

and 

K=  g  -j-  j  'b  =  orimary  admittance  per  circuit  =  —  . 

We  thus  have, 

ge  =  magnetic  energy  current,  ge*  =  loss  of  power  oy  hysteresis 
(and  eddy  currents)  per  primary  coil. 

Hence 


=  total  loss  of  energy  by  hysteresis  and  eddys, 

as  calculated  according  to  Chapter  X. 
be  =  magnetizing  current,  and 
n0be  =  effective  M.M.F.  per  primary  circuit; 

hence  ^n0be  =  total  effective  M.M.F.  ; 

z 

and 

l^-n^be  =  total  maximum  M.M.F.,  as  resultant  of  the  M.M.Fs. 
of  the  /0-phases,  combined  by  the  parallelogram  of 
M.M.Fs.* 

If   (R  =  reluctance  of  magnetic  circuit  per  pole,  as  dis- 
cussed in  Chapter  X.,  it  is 

A^^ft*. 

*  Complete  discussion  hereof,  see  Chapter  XXV. 


INDUCTION  MOTOR.  241 

Thus,  from  the  hysteretic  loss,  and  the  reluctance,  the 
constants,  g  and  b,  and  thus  the  admittance,  Fare  derived. 

Let  rQ     =  resistance  per  primary  circuit ; 
XQ    =  reactance  per  primary  circuit ; 
thus, 

•^o   =  ro  — j  XQ  =  impedance  per  primary  circuit; 

rv  =  resistance  per  secondary  circuit  reduced  to  pri- 
mary system ; 

xv  =  reactance  per  secondary  circuit  reduced  to  primary 
system,  at  full  frequency,  .A7"; 

hence, 

sx!  =  reactance  per  secondary  circuit  at  slip  s; 
and 

=  secondary  internal  impedance. 


154.   We  now  have, 
Primary  induced  E.M.F., 

E  =  -e. 
Secondary  induced  E.M.F., 

Hence, 
Secondary  current, 

*-$— 


Component  of   primary  current,   corresponding   thereto, 
primary  load  current, 

7"  --/,  = 


Primary  exciting  current, 

/0   =eY=e(g+jfy;  hence, 


242  ALTERNATING-CURRENT  PHENOMENA. 

Total  primary  current, 


E.M.F.  consumed  by  primary  impedance, 


E.M.F.  required  to  overcome  the  primary  induced  E.M.F., 

-  E  =  e; 
hence, 

Primary  terminal  voltage, 
E.  =  e  +  Ez 


We  get  thus,  in  an  induction  motor,  at  slip  s  and  active 
E.M.F.  e, 

Primary  terminal  voltage, 


Primary  current, 


or,  in  complex  expression, 
Primary  terminal  voltage, 


Primary  current, 


INDUCTION  MOTOR.  243 

To  eliminate  e,  we  divide,  and  get, 

Primary  current,  at  slip  s,  and  impressed  E.M.F.,  £0; 

f=^— 


or, 

/=  _  j  +  (>i-yji          _  E 

"  ( 


Neglecting,    in    the    denominator,    the    small    quantity 
F,  it  is 

Z,  F 


0  +  r\ 


or,  expanded, 

[(j^  +  A'0)  +  r^  -f  s^  (rog  - 

+/  [J3  (jfo+^O  +  r^+JT!  (xtg+r^+fx^  (xj>+  xj- 


Hence,    displacement    of    phase    between    current   and 
E.M.F., 
tan  ,    =  ^(^o+^ 


Neglecting  the  exciting  current,  /<„  altogether,  that  is, 
setting  Y  =  0, 
We  have 

7=  sEn^- 


„  S 

tan  <D0  = 


244 


AL  TEKNA  TING-CURRENT  PHENOMENA. 


155.  In  graphic  representation,  the  induction  motor  dia- 
gram appears  as  follows  :  — 

Denoting  the  magnetism  by  the  vertical  vector  O<b  in 
Fig.  114,  the  M.M.F.  in  ampere-turns  per  circuit  is  repre- 
sented by  vector  OF,  leading  the  magnetism  O<&  by  the 
angle  of  hysteretic  advance  a.  The  E.M.F.  induced  in  the 
secondary  is  proportional  to  the  slip  s,  and  represented  by 
~OEl  at  the  amplitude  of  180°.  Dividing  ~OEl  by  a  in  the 
proportion  of  rt  -*-  sxv  and  connecting  a  with  the  middle  b 
of  the  upper  arc  of  the  circle  OEV  this  line  intersects  the 
lower  arc  of  the  circle  at  the  point  7X  rr  Thus,  OIj\  is  the 
E.M.F.  consumed  by  the  secondary  resistance,  and  OI^ 
equal  and  parallel  to  EJ^  is  the  E.M.F.  consumed  by  the 
secondary  reactance.  The  angle,  E^OI^\  =  ^  is  the  angle 
of  secondary  lag. 


\ 


The  secondary  M.M.F.  OGl  is  in  the  direction  of  the 
vector  OIfv  Completing  the  parallelogram  of  M.M.Fs. 
with  OF  as  diagonal  and  OGl  as  one  side,  gives  the  primary 
M.M.F.  OG  as  other  side.  The  primary  current  and  the 
E.M.F.  consumed  by  the  primary  resistance,  represented  by 
OIry  is  in  line  with  OG,  the  E.M.F.  consumed  by  the  pri- 
mary reactance  90°  ahead  of  OG,  and  represented  by  OIxv 
and  their  resultant  Ofz0  is  the  E.M.F.  consumed  by  the 


INDUCTION  MOTOR. 


245 


primary  impedance.  The  E.M.F.  induced  in  the  primary 
circuit  is  OE',  and  the  E.M.F.  required  to  overcome  this 
counter  E.M.F.  is  OE  equal  and  opposite  to  OE1.  Com- 
bining OE  with  OIzQ  gives  the  primary  terminal  voltage 
represented  by  vector  OEy  and  the  angle  of  primary  lag, 
EOG 


Fig.   115. 


156.  Thus  far  the  diagram  is  essentially  the  same  as 
the  diagram  of  the  stationary  alternating-current  trans- 
former. Regarding  dependence  upon  the  slip  of  the  motor, 
the  locus  of  the  different  quantities  for  different  values  of 
the  slip  s  is  determined  thus, 


246  ALTERNATING-CURRENT  PHENOMENA. 

Let  £l  =  s£f 

Assume  in  opposition  to  O&,  a  point  A,  such  that 


O  A  -r-  7X  rx  =  Ev  -*•  /!  J.*!,  then 

/ir,  x  .£",       /ir,  x  sE      r,  _, 

=  -  ^  =  constant. 


That  is,  /^  lies  on  a  half-circle  with  OA  =  —  E'  as 
diameter. 

That  means  Gl  lies  on  a  half-circle  ^  in  Fig.  115  with 
OC  as  diameter.  In  consequence  hereof,  G0  lies  on  half- 
circle^  with  FB  equal  and  parallel  to  OCas  diameter. 

Thus  Ir0  lies  on  a  half  -circle  with  DH  as  diameter,  which 
circle  is  perspective  to  the  circle  FB,  and  Ix0  lies  on  a  half- 
circle  with  IK  as  diameter,  and  IzQ  on  a  half-circle  with  LN 
as  diameter,  which  circle  is  derived  by  the  combination  of 
the  circles  Ir0  and  Ixv 

The  primary  terminal  voltage  EQ  lies  thus  on  a  half- 
circle  e0  equal  to  the  half-circle  Iz9  and  having  to  point 
E  the  same  relative  position  as  the  half-circle  Iz^  has  to 
point  0. 

This  diagram  corresponds  to  constant  intensity  of  the 
maximum  magnetism,  O®.  If  the  primary  impressed  volt- 
age EQ  is  kept  constant,  the  circle  e0  of  the  primary  im- 
pressed voltage  changes  to  an  arc  with  O  as  center,  and  all 
the  corresponding  points  of  the  other  circles  have  to  be 
reduced  in  accordance  herewith,  thus  giving  as  locus  of  the 
other  quantities  curves  of  higher  order  which  most  con- 
veniently are  constructed  point  for  point  by  reduction  from 
the  circle  of  the  loci  in  Fig.  115. 

Torque  and  Power. 

157.  The  torque  developed  per  pole  by  an  electric  motor 
equals  the  product  of  effective  magnetism,  ®  /  V2,  times  ef- 
fective armature  M.M.F.,  F  /  V2,  times  the  sine  of  the 
angle  between  both, 


INDUCTION  MOTOR.  247 


If  «!  =  number  of  turns,  7t  =  current,  per  circuit,  with 
/rarmature  circuits,  the  total  maximum  current  polarization, 
or  M.M.F.  of  the  armature,  is 


Hence  the  torque  per  pole, 


If  q  =  the  number  of  poles  of  the  motor,  the  total  torque 
of  the  motor  is, 


The  secondary  induced  E.M.F.,  Ev  lags  90°  behind  the 
inducing  magnetism,  hence  reaches  a  maximum  displaced  in 
space  by  90°  from  the  position  of  maximum  magnetization. 
Thus,  if  the  secondary  current,  Iv  lags  behind  its  E.M.F., 
Ev  by  angle,  <av  the  space  displacement  between  armature 
current  and  field  magnetism  is 


hence  sin  (4>  fj)  =  cos  o^ 

We  have,  however, 


thus,  «!  <$ 

substituting  these  values  in  the  equation  of  the  torque,  it  is 
T. 


248  ALTERNATING-CURRENT  PHENOMENA. 

or,  in  practical  (C.G.S.)  units, 


is  the  Torque  of  the  Induction  Motor. 

At  the  slip  s,  the  frequency  N,  and  the  number  of  poles 
q,  the  linear  speed  at  unit  radius  is 


hence  the  output  of  the  motor, 
P=  TV 
or,  substituted, 


is  the  Power  of  the  Induction  Motor. 

158.  We  can  arrive  at  the  same  results  in  a  different 
way  : 

By  the  counter  E.M.F.  e  of  the  primary  circuit  with 
current  /  '  =  f0  +  7X  the  power  is  consumed,  e  I  =  e  I0  +  e  7r 
The  power  e  I0  is  that  consumed  by  the  primary  hysteresis 
and  eddys.  The  power  e  1^  disappears  in  the  primary  circuit 
by  being  transmitted  to  the  secondary  system. 

Thus  the  total  power  impressed  upon  the  .secondary 
system,  per  circuit,  is 

Pi-tf, 

Of  this  power  a  part,  £1fl,  is  consumed  in  the  secondary 
circuit  by  resistance.  The  remainder, 

P'  =  fl(e-£1), 

disappears  as  electrical  power  altogether  ;  hence,  by  the  law 
of  conservation  of  energy,  must  reappear  as  some  other 
form  of  energy,  in  this  case  as  mechanical  power,  or  as  the 
output  of  the  motor  (including  friction). 

Thus  the  mechanical  output  per  motor  circuit  is 


INDUCTION  MOTOR.  249 

Substituting, 


se; 
se 


it  is 


hence,  since  the  imaginary  part  has  no  meaning  as  power, 


and  the  total  power  of  the  motor, 

At  the  linear  speed, 
at  unit  radius  the  torque  is 


In  the  foregoing,  we  found 

£0  =  e\  1  +  j|?  +  Z,  Y 
or,  approximately, 


or, 
expanded, 


or,  eliminating  imaginary  quantities, 


250  ALTERNATING-CURRENT  PHENOMENA. 

Substituting  this  value  in  the  equations  of  torque  and  of 
power,  they  become, 

torque,  T  = 


Maximum   Torque. 

159.      The  torque  of  the  induction  motor  is  a  maximum 
for  that  value  of  slip  s,  where 


qpi  r^  Eg  s 
or,  since         T  =  -.  —  .T,  . 

4  7T  JV^   (>1 


for, 

ds 


expanded,  this  gives, 

r2 
"7 

or,  st  = 


Substituting  this  in  the  equation  of  torque,  we  get  the 
value  of  maximum  torque, 


That  is,  independent  of  the  secondary  resistance,  rr 
The  power  corresponding  hereto  is,  by  substitution  of  st 
in  P, 

Pt  = ; 


This  power  is  not  the  maximum  output  of  the  motor, 
but  already  below  the  maximum  output.  The  maximum 
output  is  found  at  a  lesser  slip,  or  higher  speed,  while  at 
the  maximum  torque  point  the  output  is  already  on  the 
decrease,  due  to  the  decrease  of  speed. 


INDUCTION  MOTOR.  251 

With  increasing  slip,  or  decreasing  speed,  the  torque  of 
the  induction  motor  increases ;  or  inversely,  with  increasing 
load,  the  speed  of  the  motor  decreases,  and  thereby  the 
torque  increases,  so  as  to  carry  the  load  down  to  the  slip  st, 
corresponding  to  the  maximum  torque.  At  this  point  of 
load  and  slip  the  torque  begins  to  decrease  again ;  that  is, 
as  soon  as  with  increasing  load,  and  thus  increasing  slip, 
the  motor  passes  the  maximum  torque  point  st,  it  "  falls  out 
of  step,"  and  comes  to  a  standstill. 

Inversely,  the  torque  of  the  motor,  when  starting  from 
rest,  will  increase  with  increasing  speed,  until  the  maximum 
torque  point  is  reached.  From  there  towards  synchronism 
the  torque  decreases  again. 

In  consequence  hereof,  the  part  of  the  torque-speed 
curve  below  the  maximum  torque  point  is  in  general  un- 
stable, and  can  be  observed  only  by  loading  the  motor 
with  an  apparatus,  whose  countertorque  increases  with  the 
speed  faster  than  the  torque  of  the  induction  motor. 

In  general,  the  maximum  torque  point,  st,  is  between 
synchronism  and  standstill,  rather  nearer  to  synchronism. 
Only  in  motors  of  very  large  armature  resistance,  that  is 
low  efficiency,  st  >  1,  that  is,  the  maximum  torque  falls 
below  standstill,  and  the  torque  constantly  increases  from 
synchronism  down  to  standstill. 

It  is  evident  that  the  position  of  the  maximum  torque 
point,  st  can  be  varied  by  varying  the  resistance  of  the 
secondary  circuit,  or  the  motor  armature.  Since  the  slip 
of  the  maximum  torque  point,  st,  is  directly  proportional  to 
the  armature  resistance,  rlf  it  follows  that  very  constant 
speed  and  high  efficiency  will  bring  the  maximum  torque 
point  near  synchronism,  and  give  small  starting  torque, 
while  good  starting  torque  means  a  maximum  torque  point 
at  low  speed ;  that  is,  a  motor  with  poor  speed  regulation* 
and  low  efficiency. 

Thus,  to  combine  high  efficiency  and  close  speed  regula- 
tion with  large  starting  torque,  the  armature  resistance  has 


252  ALTERNATING-CURRENT  PHENOMENA. 

to  be  varied  during  the  operation  of  the  motor,  and  the 
motor  started  with  high  armature  resistance,  and  with  in- 
creasing speed  this  armature  resistance  cut  out  as  far  as 
possible. 

160.  If  *=:1,__ 

it  is  ^  =  Vr02  +  (xl  +  *0)2. 

In  this  case  the  motor  starts  with  maximum  torque,  and 
when  overloaded  does  not  drop  out  of  step,  but  gradually 
slows  down  more  and  more,  until  it  comes  to  rest. 

If,  st>l, 

then  ^  >  Vr02  +  (^  +  *0)2. 

In  this  case,  the  maximum  torque  point  is  reached  only 
by  driving  the  motor  backwards,  as  countertorque. 

As  seen  above,  the  maximum  torque  Tt,  is  entirely  in- 
dependent of  the  armature  resistance,  and  likewise  is  the 
current  corresponding  thereto,  independent  of  the  armature 
resistance.  Only  the  speed  of  the  motor  depends  upon  the 
armature  resistance. 

Hence  the  insertion  of  resistance  into  the  motor  arma- 
ture does  not  change  the  maximum  torque,  and  the  current 
corresponding  thereto,  but  merely  lowers  the  speed  at  which 
the  maximum  torque  is  reached. 

The  effect  of  resistance  inserted  into  the  induction  motor 
is  merely  to  consume  the  E.M.F.,  which  otherwise  would 
find  its  mechanical  equivalent  in  an  increased  speed,  analo- 
gous as  resistance  in  the  armature  circuit  of  a  continuous- 
current  shunt  motor. 

Further  discussion  on  the  effect  of  armature  resistance 
is  found  under  "  Starting  Torque." 

Maximum  Power. 

161.  The  power  of  an  induction  motor  is  a  maximum 
for  that  slip,  sv,  where 


INDUCTION  MOTOR.  253 


expanded,  this  gives 

sn  —  - 


substituted  in  P,  we  get  the  maximum  power, 


2  {('i  +  ''o)  +      (^  +  r0)2  +  (^i  +  *o)2} 

This  result  has  a  simple  physical  meaning  :  (i\  +  r0)  =  r 
is  the  total  resistance  of  the  motor,  primary  plus  secondary 
(the  latter  reduced  to  the  primary),  (x^  +  x^  is  the  total 
reactance,  and  thus  Vrx  +  r0)2  +  (x^  +  x0}z  =  z  is  the  total 
impedance  of  the  motor.  Hence 


is  the  maximum  output  of  the  induction  motor,  at  the  slip, 


The  same  value  has  been  derived  in  Chapter  IX.,  as  the 
maximum  power  which  can  be  transmitted  into  a  non- 
inductive  receiver  circuit  over  a  line  of  resistance  r,  and 
impedance  z,  or  as  the  maximum  output  of  a  generator,  or 
of  a  stationary  transformer.  Hence  : 

The  maximum  output  of  an  induction  motor  is  expressed 
by  the  same  formula  as  the  maximum  output  of  a  generator, 
or  of  a  stationary  transformer,  or  the  maximum  output  which 
can  be  transmitted  over  an  inductive  line  into  a  non-inductive- 
receiver  circuit. 

The  torque  corresponding  to  the  maximum  output  Pp  is,. 


254  ALTERNATING-CURRENT  PHENOMENA. 

This   is   not   the  maximum  torque  ;  but    the  maximum 
torque,  Tt,  takes  place  at  a  lower  speed,  that  is,  greater  slip, 


•  since, 


-that  is,  st  >  sp. 

It  is  obvious  from  these  equations,  that,  to  reach  as  large 
an  output  as  possible,  r  and  z  should  be  as  small  as  possible  ; 
that  is,  the  resistances  ^  +  r0,  and  the  impedances,  z, 
and  thus  the  reactances,  x±  +  x0,  should  be  small.  Since 
r±  +  r0  is  usually  small  compared  with  x^  -f-  x0  it  follows,  that 
the  problem  of  induction  motor  design  consists  in  con- 
structing the  motor  so  as  to  give  the  minimum  possible 
reactances,  x^  +  x0. 

Starting  Torque. 

162.  In  the  moment  of  starting  an  induction  motor, 
the  slip  is 

hence,  starting  current, 


Oo  - 

or,  expanded,  with  the  rejection  of   the  last  term   in  the 
denominator,  as  insignificant, 


T  _io11       010,io1          . 
-  8 


and,  displacement  of  phase,  or  angle  of  lag, 

fi  +  r0]  +  *!  [Jfx  4-  Jf0])  -  jf  (r0  ^  -  *0  rt) 


„    _ 
1  W° 


r0) 


INDUCTION  MOTOR.  255 

Neglecting  the  exciting  current,  g  =  0  =  b,  these  equa- 
tions assume  the  form, 


or,  eliminating  imaginary  quantities, 


and  tan  w0  = 


+  'o 


That  means,  that  in  starting  the  induction  motor  without 
additional  resistance  in  the  armature  circuit,  —  in  which  case 
^  +  x0  is  large  compared  with  t\  •+•  r0,  and  the  total  impe- 
dance, z,  small,  —  the  motor  takes  excessive  and  greatly 
lagging  currents. 

The  starting  torque  is 


T0= 


That  is,  the  starting  torque  is  proportional  to  the 
armature  resistance,  and  inversely  proportional  to  the  square 
of  the  total  impedance  of  the  motor. 

It  is  obvious  thus,  that,  to  secure  large  starting  torque, 
the  impedance  should  be  as  small,  and  the  armature  resis- 
tance as  large,  as  possible.  The  former  condition  is  the 
condition  of  large  maximum  output  and  good  efficiency 
and  speed  regulation  ;  the  latter  condition,  however,  means 
inefficiency  and  poor  regulation,  and  thus  cannot  properly 
be  fulfilled  by  the  internal  resistance  of  the  motor,  but  only 
by  an  additional  resistance  which  is  short-circuited  while 
the  motor  is  in  operation. 


256  ALTERNATING-CURRENT  PHENOMENA. 

Since,  necessarily, 

ri<*, 

''<•< 


and  since  the  starting  current  is,  approximately, 

7    =f  , 
we  have,  Ta  < 


would  be  the  theoretical  torque  developed  at  100  per  cent 
efficiency  and  power  factor,  by  E.M.F.,  E0,  and  current,  /, 
at  synchronous  speed. 

Thus,  T0<T00, 

and  the  ratio  between  the  starting  torque  T0,  and  the  theo- 
retical maximum  torque,  T^,  gives  a  means  to  judge  the 
perfection  of  a  motor  regarding  its  starting  torque. 

This  ratio,  T0  /  Tw,  exceeds  .9  in  the  best  motors. 

Substituting  7  =  E0  /  z  in  the  equation  of  starting  torque, 
it  assumes  the  form, 

7V,. 


Since  4  IT  N /  q  =  synchronous  speed,  it  is  : 

The  starting  torque  of  the  induction  motor  is  equal  to  the 
resistance  loss  in  the  motor  armature,  divided  by  the  synchro- 
nous speed. 

The  armature  resistance  which  gives  maximum  starting 
torque  is 


INDUCTION  MOTOR.  257 


dr, 
expanded,  this  gives, 


the  same  value  as  derived  in  the  paragraph  on  "maximum 
torque." 

Thus,  adding  to  the  internal  armature  resistance,  r/  in 
starting  the  additional  resistance, 


makes  the  motor  start  with  maximum  torque,  while  with  in- 
creasing speed  the  torque  constantly  decreases,  and  reaches 
zero  at  synchronism.  Under  these  conditions,  the  induc- 
tion motor  behaves  similarly  to  the  continuous-current  series 
motor,  varying  in  the  speed  with  the  load,  the  difference 
being,  however,  that  the  induction  motor  approaches  a 
definite  speed  at  no  load,  while  with  the  series  motor  the 
speed  indefinitely  increases  with  decreasing  load. 

The  additional  armature  resistance,  t\",  required  to  give 
a  certain  starting  torque,  if  found  from  the  equation  of 
starting  torque : 

Denoting  the  internal  armature  resistance  by  rj,  the  total 
armature  resistance  is  ^  =  r^  +  r". 

and  thus,  ?A  Eg rj  +  r" 

4  TT  N  (r^  +  r^  +  r0)2  +  (Xl  +  *0)2 ' 
hence, 


This  gives  two  values,  one  above,  the  other  below,  the 
maximum  torque  point. 


258  ALTERNATING-CURRENT  PHENOMENA. 

Choosing  the  positive  sign  of  the  root,  we  get  a  larger 
armature  resistance,  a  small  current  in  starting,  but  the 
torque  constantly  decreases  with  the  speed. 

Choosing  the  negative  sign,  we  get  a  smaller  resistance, 
a  large  starting  current,  and  with  increasing  speed  the 
torque  first  increases,  reaches  a  maximum,  and  then  de- 
creases again  towards  synchronism. 

These  two  points  correspond  to  the  two  points  of  the 
speed-torque  curve  of  the  induction  motor,  in  Fig.  116, 
giving  the  desired  torque  T0. 

The  smaller  value  of  r1"  will  give  fairly  good  speed  regu- 
lation, and  thus  in  small  motors,  where  the  comparatively 
large  starting  current  is  no  objection,  the  permanent  arma- 
ture resistance  may  be  chosen  to  represent  this  value. 

The  larger  value  of  rj'  allows  to  start  with  minimum 
current,  but  requires  cutting  out  of  the  resistance  after  the 
start,  to  secure  speed  regulation  and  efficiency. 

Synchronism. 
163.    At  synchronism,  s  =  0,  we  have, 


or, 


0,  T=Q; 


that  is,  power  and  torque  are  zero.  Hence,  the  induction 
motor  can  never  reach  complete  synchronism,  but  must 
slip  sufficiently  to  give  the  torque  consumed  by  friction. 

Running  near  Synchronism. 

164.  When  running  near  synchronism,  at  a  slip  s  above 
the  maximum  output  point,  where  s  is  small,  from  .02  to 
.05  at  full  load,  the  equations  can  be  simplified  by  neglect- 
ing terms  with  s,  as  of  higher  order. 


INDUCTION  MOTOR.  25  £ 

We  then  have,  current, 


or,  eliminating  imaginary  quantities, 


angle  of  lag,  o*i  +  *o  , 

c2  (r_  -I-  <r_\  -4-  r.2  h  r. 

tan  w0 

T  = 

or,  inversely, 


A  A 


that  is, 

Near  sychronism,  the  slip,  s,  of  an  induction  motor,  or 
its  drop  in  speed,  is  proportional  to  the  armature  resistance> 
i\  and  to  the  power,  P,  or  torque,  T. 

Example. 

165.  As  an  instance  are  shown,  in  Fig.  116,  character- 
istic curves  of  a  20  horse-power  three-phase  induction  motor, 
of  900  revolutions  synchronous  speed,  8  poles,  frequency 
of  60  cycles. 

The  impressed  E.M.F.  is  110  volts  between  lines,  and 
the  motor  star  connected,  hence  the  E.M.F.  impressed  per 
circuit  : 

~  =  63.5  ;  or  EQ  =  63.5. 


260 


AL  TERN  A  TING-CURRENT  PHENOMENA. 


The  constants  of  the  motor  are  : 

Primary  admittance,  Y  =  .1  +  .4  j. 
Primary  impedance,  Z  =  .03  —  .09  j. 
Secondary  impedance,  Zx  =  .02  —  .085/. 

In   Fig.   116  is  shown,  with  the  speed  in  per  cent  of 

•synchronism,   as  abscissae,   the    torque  in   kilogrammetres, 

as  ordinates,   in  drawn  lines,   for  the  values  of  armature 
resistance : 


116.    Speed  Characteristics  of  Induction  Motor. 


rt  =  .02    :  short  circuit  of  armature,  full  speed. 

^  =  .045  :  .025  ohms  additional  resistance. 

^  =  .18    :  .16  ohms  additional,  maximum  starting  torque. 

^  =  .75    :  .73  ohms  additional,  same  starting  torque  as  rt  ==  .045. 

On  the  same  Figure  is  shown  the  current  per  line,  in 
dotted  lines,  with  the  verticals  or  torque  as  abscissae,  and 
the  horizontals  or  amperes  as  ordinates.  To  the  same 
torque  always  corresponds  the  same  current,  no  matter 
what  the  speed  be. 


INDUCTION  MOTOR. 


261 


On  Fig.  117  is  shown,  with  the  current  input  per  line  as 
abscissae,  the  torque  in  kilogrammetres  and  the  output  in 
horse-power  as  ordinates  in  drawn  lines,  and  the  speed  and 
the  magnetism,  in  per  cent  of  their  synchronous  values,  as 
ordinates  in  dotted  lines,  for  the  armature  resistance  ^  =  .02 
or  short  circuit. 


20 


lase  Induotio     Motor. 


.    60Cyc 


110V 


Jiagram 


=.03-.09j 
z£0=J&B 


\ 


\\ 


\\ 


12 

-1 


Amperes 
150 1  200 


2,50 


300 


Fig.  117.    Current  Characteristics  of  Induction  Motor. 

In  Fig.  118  is  shown,  with  the  speed,  in  per  cent  of 
synchronism,  as  abscissae,  the  torque  in  drawn  line,  and 
the  output  in  dotted  line,  for  the  value  of  armature  resist- 
ance ?i  =  .045,  for  the  whole  range  of  speed  from  120  per 


262 


ALTERNA TING-CURRENT  PHENOMENA. 


cent  backwards  speed  to  220  per  cent  beyond  synchronism, 
showing  the  two  maxima,  the  motor  maximum  at  s  =  .25, 
and  the  generator  maximum  at  s  =  —  .25. 

166.     As  seen  in  the  preceding,  the  induction  motor  is 
characterized  by  the  three  complex  imaginary  constants, 

Y0  =  g0  +jbw  the  primary  exciting  admittance, 
Z0  =  r0  —jx0,  the  primary  self-inductive  impedance,  and 
Zi  =  r±  —  jx^  the  secondary  self-inductive  impedance, 


Fig.  1 18.    Speed  Characteristics  of  Induction  Motor. 

reduced  to  the  primary  by  the  ratio  of  secondary  to  pri- 
mary turns. 

From  these  constants  and  the  impressed  E.M.F.  cot  the 
motor  can  be  calculated  as  follows  : 

Let, 

e  =  counter  E.M.F.  of  motor,  that  is  E.M.F.  induced  in 
the  primary  by  the  mutual  magnetic  flux. 

At  the  slip  s  the  E.M.F.  induced  in  the  secondary  cir- 
cuit is,  se 


INDUCTION  MOTOR.  263 


Thus  the  secondary  current, 


where, 


«l  =  -5T 


r*  +  Atf  r?  + 

The  primary  exciting  current  is, 


thus,  the  total  primary  current, 

/0  =  /!  +  /oo  =  *  (^i  +  A) 
where, 


The  E.M.F.  consumed  by  the  primary  impedance  is, 
^  =  /oZ0  =  *  (r0  ->0)  (^ 


the  primary  counter  E.M.F.  is  e,  thus  the  primary  impressed 
E.M.F., 

£, 
where, 

c\  — 
or,  absolute, 

^0   = 

hence, 


This  value  substituted  gives, 

Secondary  current, 

ffi+A 
A  =  *b  T7= 


Primary  current, 

°~ 

Impressed  E.M.F., 


264  ALTERNATING-CURRENT  PHENOMENA. 

Thus  torque,  in   synchronous  watts  (that   is,  the  watts 
output  the  torque  would  produce  at  synchronous  speed), 


tf  +  tf 

hence,  the  torque  in  absolute  units, 


=       = 


N      (f*  +  r22)  W 
where  N=  frequency. 

The  power  output  is  torque  times  speed,  thus  : 


The  power  input  is, 


^•l2  + 

The  voltampere  input, 


o2  (  Vi  +  V,)        /o2  (  Vi  -  V8) 


hence, 

efficiency, 

J\  _  a,  (I  -  s) 

J?      Vi  +  V2 

power  factor, 


apparent  efficiency, 


<2o 

torque  efficiency,  * 
a. 


./V      Vi  +  V. 

*  That  5s  the  ratio  of  actual  torque  to  torque  which  would  be  profloced,  if  there  were  nc 
losses  of  energy  in  the  motor,  at  the  same  power  input. 


INDUCTION  MOTOR.  265 


apparent  torque  efficiency,* 

rrt 

~Q0  ~  V  W~+1?YT^ 


167.  Most  instructive  in  showing  the  behavior  of  an 
induction  motor  are  the  load  curves  and  the  speed  curves. 

The  load  curves  are  curves  giving,  with  the  power  out- 
put as  abscissae,  the  current  imput,  speed,  torque,  power 
factor,  efficiency,  and  apparent  efficiency,  as  ordinates. 

The  speed  curves  give,  with  the  speed  as  abscissae,  the 
torque,  current  input,  power  factor,  torque  efficiency,  and 
apparent  torque  efficiency,  as  ordinates. 

The  load  curves  characterize  the  motor  especially  at  its 
normal  running  speeds  near  synchronism,  the  speed  curves 
over  the  whole  range  of  speed. 

In  Fig.  119  are  shown  the  load  curves,  and  in  Fig.  120 
the  speed  curves  of  a  motor  of  the  constants, 
K0  =  .01  +  .!/ 

z*  =  .i  -.3> 

Z,  =  .1    -  .3j 

INDUCTION  GENERATOR. 

168.  In  the  foregoing,  the  range  of  speed  from  s  =  1, 
standstill,  to  s  =  0,  synchronism,  has  been  discussed.     In 
this  range  the  motor  does  mechanical  work. 

It  consumes  mechanical  power,  that  is,  acts  as  generator 
or  as  brake  outside  of  this  range. 

For,  s  >  1,  backwards  driving,  P  becomes  negative, 
representing  consumption  of  power,  while  T  remains  posi- 
tive ;  hence,  since  the  direction  of  rotation  has  changed, 
represents  consumption  of  power  also.  All  this  power  is 
consumed  in  the  motor,  which  thus  acts  as  brake. 

For,  s  <  0,  or  negative,  P  and  T  become  negative,  and 
the  machine  becomes  an  electric  generator,  converting  me- 
chanical into  electric  energy. 

*  That  is  the  ratio  of  actual  torque  to  torque  which  would  be  produced  if  there  were 
neither  losses  of  energy  nor  phase  displacement  in  the  motor,  at  the  same  voltampere  input. 


266 


ALTERNA  TING-CURRENT  PHENOMENA. 


The  calculation  of  the  induction  generator  at  constant 
frequency,  that  is,  at  a  speed  increasing  with  the  load  by  the 
negative  slip,  slt  is  the  same  as  that  of  the  induction  motor 
except  that  sl  has  negative  values,  and  the  load  curves  for 
the  machine  shown  as  motor  in  Fig.  119  are  shown  in  Fig. 
121  for  negative  slip  s{  as  induction  generator. 


CURV 


POWER 
4000 


"£> 


Fig.  119. 


Again,  a  maximum  torque  point  and  a  maximum  output 
point  are  found,  and  the  torque  and  power  increase  from 
zero  at  synchronism  up  to  a  maximum  point,  and  then  de- 
crease again,  while  the  current  constantly  increases. 


INDUCTION  MOTOR. 


267 


Fig.   120. 


268  ALTERNATING-CURRENT  PHENOMENA. 

169.  The  induction  generator  differs  essentially  from 
the  ordinary  synchronous  alternator  in  so  far  as  the  induc- 
tion generator  has  a  definite  power  factor,  while  the  syn- 
chronous alternator  has  not.  That  is,  in  the  synchronous 
alternator  the  phase  relation  between  current  and  terminal 
voltage  entirely  depends  upon  the  condition  of  the  external 
circuit.  The  induction  generator,  however,  can  operate 
only  if  the  phase  relation  of  current  and  E.M.F.,  that  is,  the 
power  factor  required  by  the  external  circuit,  exactly  coin- 
cides with  the  internal  power  factor  of  the  induction  gen- 
erator. This  requires  that  the  power  factor  either  of  the 
external  circuit  or  of  the  induction  generator  varies  with 
the  voltage,  so  as  to  permit  the  generator  and  the  external 
circuit  to  adjust  themselves  to  equality  of  power  factor. 

Beyond  magnetic  saturation  the  power  factor  decreases  ; 
that  is,  the  lead  of  current  increases  in  the  induction  ma- 
chine. Thus,  when  connected  to  an  external  circuit  of  con- 
stant power  factor  the  induction  generator  will  either  not 
generate  at  all,  if  its  power  factor  is  lower  than  that  of  the 
external  circuit,  or,  if  its  power  factor  is  higher  than  that  of 
the  external  circuit,  the  voltage  will  rise  until  by  magnetic 
saturation  in  the  induction  generator  its  power  factor  has 
fallen  to  equality  with  that  of  the  external  circuit.  This, 
however,  requires  magnetic  saturation  in  the  induction  gen- 
erator, which  is  objectionable,  due  to  excessive  hysteresis 
losses  in  the  alternating  field. 

To  operate  below  saturation, —  that  is,  at  constant  inter- 
nal power  factor, — the  induction  generator  requires  an  exter- 
nal circuit  with  leading  current,  whose  power  factor  varies 
with  the  voltage,  as  a  circuit  containing  synchronous  motors 
or  synchronous  converters.  In  such  a  circuit,  the  voltage 
of  the  induction  generator  remains  just  as  much  below  the 
counter  E.M.F.  of  the  synchronous  motor  as  necessary  to 
give  the  required  leading  exciting  current  of  the  induction 
generator,  and  the  synchronous  motor  can  thus  to  a  certain 
extent  be  called  the  exciter  of  the  induction  generator. 


INDUCTION  MOTOR.  269 

When  operating  self-exciting,  that  is  shunt-wound,  con- 
verters from  the  induction  generator,  below  saturation  of 
both  the  converter  and  the  induction  generator,  the  condi- 
tions are  unstable  also,  and  the  voltage  of  one  of  the  two 
machines  must  rise  beyond  saturation  of  its  magnetic  field. 

When  operating  in  parallel  with  synchronous  alternat- 
ing generators,  the  induction  generator  obviously  takes  its 
leading  exciting  current  from  the  synchronous  alternator, 
which  thus  carries  a  lagging  wattless  current. 

170.  To  generate  constant  frequency,  the  speed  of  the 
induction  generator  must  increase  with  the  load.  Inversely, 
when  driven  at  constant  speed,  with  increasing  load  on  the 
induction  generator,  the  frequency  of  the  current  generated 
thereby  decreases.  Thus,  when  calculating  the  character- 
istic curves  of  the  constant  speed  induction  generator,  due 
regard  has  to  be  taken  of  the  decrease  of  frequency  with 
increase  of  load,  or  what  may  be  called  the  slip  of  fre- 
quency, s. 

Let  in  an  induction  generator, 

Y0  =  gQ  +  j\  —  primary  exciting  admittance, 

Z0  =  r0  —  jxQ  =  primary  self-inductive  impedance, 

Zi  =  r^  —  jXj_  =  secondary  self-inductive  impedance, 

reduced  to  primary,  all  these  quantities  being  reduced  to 
the  frequency  of  synchronism  with  the  speed  of  the  ma- 
chine, N. 

Let  e  —  induced  E.M.F.,  reduced  to  full  frequency. 

s  =  slip  of  frequency,  thus  :  (1-j)  N  =  frequency  gener- 
ated by  machine. 

We  then  have 

Secondary  induced  E.M.F. 
se 
thus,  secondary  current, 


r          in 
r\  —  Jsx\ 


270  ALTERNATING-CURRENT  PHENOMENA. 

where, 


primary  exciting  current, 

In  =  EY0  =  e 
thus,  total  primary  current, 

/0  =  /i  +  foo 
where, 

^1   =   <*\   +  £b 

primary  impedance  voltage, 
&  =  S0(r0- 

primary  induced  E.M.F., 


thus,  primary  terminal  voltage, 

£0  =  e(l-s)  -S0(r0-j[l-  s]  x0)  =  e 
where, 

fi  =  !  -  s  ~  rA  -  (1  -  s 
hence,  absolute, 

e0  =  e  V^ 
and, 


Thus, 

Secondary  current, 

T  eO  (ai 


Primary  current, 

j  _  eo  (A  + A) 

Primary  terminal  voltage, 

j-.  ^0  \^"l 

£«  =  —T-, 


INDUCTION  MOTOR. 
Torque  and  mechanical  power  input, 

T—  P  —\f  nl  —    e°ai 
r*  ~  \-e  ^  ~  7^+^ 

Electrical  output, 


271 


ELECTRICAL   OUTPUT      P    ,    WATTS 
1000  2000  3COO  4000  fiOOO  fiOOO  7000  8000 


Fig.  122. 


Voltampere  output, 
G,  =  < 

Efficiency, 

j 

power  factor, 


272  AL  TERNA  TING-CURRENT  PHENOMENA. 

or, 

p,j     b*  -  V, 
=  ^-  =  ^T^ 

In  Fig.  122  is  plotted  the  load  characteristic  of  a  con- 
stant speed  induction  generator,  at  constant  terminal  vol- 
tage e  0  =  110,  and  the  constants, 

K0  =  .01  +  .!/ 


171.  As  instance  may  be  considered  a  power  trans- 
mission from  an  induction  generator  of  constants  Y0,  Z0, 
Zj,  over  a  line  of  impedance  Z  =  r  —jx,  into  a  synchron- 
ous motor  of  synchronous  impedance  Zz  =  rz  —  jxz,  operat- 
ing at  constant  field  excitation. 

Let,  e0  =  counter  E.M.F.  or  nominal  induced  E.M.F.  of 
synchronous  motor  at  full  frequency  ;  that  is,  frequency  of 
synchronism  with  the  speed  of  the  induction  generator. 
By  the  preceding  paragraph  the  primary  current  of  the 
induction  generator  was, 


primary  terminal  voltage, 
E0  =  e 
thus,  terminal  voltage  at  synchronous  motor  terminals, 


where, 

4  =  fi  ~  rA  ~  C1  -  J)  *A         4  = 

Counter  E.M.F.  of  synchronous  motor, 

E2 

' 

where, 

/  =  4  -  r&  -  (1 
or  absolute, 


INDUCTION  MOTOR. 


since,  however, 


Z=.0|4-6j 

ULL  F  EQUE 
EXCIT/ 
5  VOL' 


OUTPUT    OF    SYNCHRONOUS,  WATTS 
1000  2000        I        8000  4000  5000 


274  ALTERNATING-CURRENT  PHENOMENA. 


Thus, 


Current,  _  e2  (1  -  j)  (^  +y7;2) 

' 


Terminal  voltage  at  induction  generator, 


Terminal  voltage  at  synchronous  motor, 


and  herefrom  in  the  usual  way  the  efficiencies,  power  fac- 
tor, etc.  are  derived. 

When  operated  from  an  induction  generator,  a  syn- 
chronous motor  gives  a  load  characteristic  very  similar  to 
that  of  an  induction  motor  operated  from  a  synchronous 
generator,  but  in  the  former  case  the  current  is  leading,  in 
the  latter  lagging. 

In  either  case,  the  speed  gradually  falls  off  with  increas- 
ing load  (in  the  synchronous  motor,  due  to  the  falling  off 
of  the  frequency  of  the  induction  generator),  up  to  a  maxi- 
mum output  point,  where  the  motor  drops  out  of  step  and 
comes  to  standstill. 

Such  a  load  characteristic  of  the  induction  generator  in 
Fig.  121,  feeding  a  synchronous  motor  of  counter  E.M.F. 
eQ  =  125  volts  (at  full  frequency)  and  synchronous  impe- 
dance Z2  =  .04  —  Gj,  over  a  line  of  negligible  impedance 
is  shown  in  Fig.  123. 

CONCATENATION,  OR  TANDEM  CONTROL  OF  INDUCTION 
MOTORS. 

172.  If  of  two  induction  motors  the  secondary  of  the 
first  motor  is  connected  to  the  primary  of  the  second  motor, 
the  second  machine  operates  as  motor  with  the  E.M.F.  and 
frequency  impressed  upon  it  by  the  secondary  of  the  first 
machine,  which  acts  as  general  alternating-current  trans- 
former, converting  a  part  of  the  primary  impressed  power 


INDUCTION  MOTOR.  275 

into  secondary  electrical  power  for  the  supply  of  the  second 
machine,  and  a  part  into  mechanical  work. 

The  frequency  of  the  secondary  E.M.F.  of  the  first  motor, 
and  thus  the  frequency  impressed  upon  the  second  motor,  is 
the  frequency  of  slip  below  complete  synchronism,  s.  The 
frequency  of  the  secondary  induced  E.M.F.  of  the  second 
motor  is  the  difference  between  its  impressed  frequency, 
s,  and  its  speed  ;  thus,  if  both  motors  are  connected  together 
mechanically  to  turn  at  the  same  speed,  1  —  s,  the  secondary 
frequency  of  the  second  motor  is  2^—1,  hence  equal  to 
zero  at  s  =  .5.  That  is,  the  second  motor  reaches  its  syn- 
chronism at  half  speed.  At  this  speed  its  torque  becomes 
equal  to  zero,  the  energy  current  flowing  into  it,  and  conse- 
quently the  energy  component  of  the  secondary  current  of 
the  first  "motor,  and  thus  the  torque  of  the  first  motor  be- 
comes equal  to  zero  also,  when  neglecting  the  hysteresis 
energy  current  of  the  second  motor.  That  is,  a  system  of 
concatenated  motors  with  short-circuited  secondary  of  the 
second  motor  approaches  half  synchronism,  in  the  same 
manner  as  the  ordinary  induction  motor  approaches  syn- 
chronism. With  increasing  load,  its  slip  below  half  syn- 
chronism increases. 

More  generally,  any  pair  of  induction  motors  connected 
in  concatenation  divide  the  speed  so  that  the  sum  of  their 
two  respective  speeds  approaches  synchronism  at  no  load  ; 
or,  still  more  generally,  any  number  of  concatenated  motors 
run  at  such  speeds  that  the  sum  of  the  speeds  approaches 
synchronism  at  no  load. 

With  mechanical  connection  between  the  two  motors, 
concatenation  thus  offers  a  means  to  operate  a  pair  of 
induction  motors  at  full  efficiency  at  half  speed  in  tandem, 
as  well  as  at  full  speed  in  parallel,  and  thus  gives  the  same 
advantage  as  the  series-parallel  control  of  the  continuous- 
current  motor. 

In  starting,  a  concatenated  system  is  controlled  by  re- 
sistance in  the  armature  of  the  second  motor. 


276  ALTERNATING-CURRENT  PHENOMENA. 

Since,  with  increasing  speed,  the  frequency  impressed 
upon  the  second  motor  decreases  proportionally  to  the  de- 
crease of  voltage,  when  neglecting  internal  losses  in  the 
first  motor,  the  magnetic  density  of  the  second  motor  re- 
mains practically  constant,  and  thus  its  torque  the  same  as 
when  operated  at  full  voltage  and  full  frequency  under  the 
same  conditions. 

At  half  synchronism  the  torque  of  the  concatenated 
couple  becomes  zero,  and  above  half  synchronism  the  sec- 
ond motor  runs  beyond  its  impressed  frequency  ;  that  is, 
becomes  generator.  In  this  case,  due  to  the  reversal  of 
current  in  the  secondary  of  the  first  motor,  its  torque 
becomes  negative  also,  that  is  the  concatenated  couple 
becomes  induction  generator  above  half  synchronism.  At 
about  two-thirds  synchronism,  with  low  resistance  armature, 
the  torque  of  the  couple  becomes  zero  again,  and  once  more 
positive  between  about  two-thirds  synchronism  and  full  syn- 
chronism, and  negative  once  more  beyond  full  synchronism. 
With  high  resistance  in  the  secondary  of  the  second  motor, 
the  second  range  of  positive  torque,  below  full  synchronism, 
disappears,  more  or  less. 

173.  The  calculation  of  a  concatenated  couple  of  in- 
duction motors  is  as  follows, 

Let 

N  =  frequency  of  main  circuit, 

s  =  slip  of  the  first  motor  from  synchronism. 

the  frequency  induced  in  the  secondary  of  the  first  motor 
and  thus  impressed  upon  the  primary  of  the  second  motor 
is,  s  N. 

The^peed  of  the  first  motor  is  (1 — s)  N,  thus  the  slip 
of  the  second  motor,  or  the  frequency  induced  in  its  sec- 
ondary, is 


INDUCTION  MOTOR.  277 

Let 

e  =  counter  E.M.F.  induced  in  the  secondary  of  the  sec- 
ond motor,  reduced  to  full  frequency. 

Z0  =  r0  —  jxQ  =  primary  self-inductive  impedance. 

Z^  =  i\  —jxv  =  secondary  self-inductance  impedance. 

Y  —  g  +jb  =  primary  exciting  admittance  of  each  mo- 
tor, all  reduced  to  full  frequency  and  to  the  primary  by  the 
ratio  of  turns. 

We  then  have, 

Second  motor, 
secondary  induced  E.M.F., 

*(*/-!) 

secondary  current, 


where, 

(2s-l)r1 


i  ~  r*+  (2J-1)2^12  z  ~  r*+  (2s- 

primary  exciting  current, 

4  =  *  (g  +JI>} 
thus,  total  primary  current, 

72  =  7,  +  70  =  e  ( 
where, 


primary  induced  E.M.F., 

se 
primary  impedance  voltage, 

ft  (ro  —  >^o) 
thus,  primary  impressed  E.M.F., 

£3  =  se  +  72  (r0  -jsx0)  =  e  (^ 
where, 


First  motor, 
secondary  current, 


278       ALTERNATING-CURRENT  PHENOMENA. 

secondary  induced  E.M.F., 

£9  = 
where, 


primary  induced  E.M.F., 

EI  =  - 
where, 

s 
primary  exciting  current, 

total  primary  current, 
where, 


primary  impedance  voltage, 

|(>o  ~> 

thus,  primary  impressed  E.M.F., 
£0  =  E,  +  S(r0  ->0 
where, 

^i  =/i  +  ^o5i  +  *b£a 

or,  absolute, 

<-„ 
and, 


V  V  +  V 

Substituting  now  this  value  of  ^  in  the  preceding  gives 
the  values  of  the  currents  and  E.M.F.'s  in  the  different 
circuits  of  the  motor  series. 

*  At  s  =  0  these  terms/i  and/s  become  indefinite,  and  thus  at  and  very  near  synchronism 
have  to  be  derived  by  substituting  the  complete  expressions  fory^  andy"2. 


INDUCTION  MOTOR.  279 

In  the  second  motor,  the  torque  is, 

T2  =  [,/J  =  ^ 
hence,  its  power  output, 

/»,=  (!-  s)  r2  =  (1  -  s)  <?ai 
The  power  input  is, 


hence,  the  efficiency, 

PS        (1  -  s)  fa, 


the  power  factor, 


etc. 

In  the  first  motor, 
the  torque  is, 


the  power  output, 

PI  =  71  (1  -  j) 

=  ^  (1  -  ,)  (/^  -h/A) 

the  power  input, 

P1  = 


Thus,  the  efficiency, 

^  (1  -  Q  (/A  +/A) 


+  ^2)  -  (^  + 
the  power  factor  of  the  whole  system, 


280  ALTERNATING-CURRENT  PHENOMENA. 

the  power  factor  of  the  first  motor, 


the  total  efficiency  of  the  system, 


etc. 


f  /ff.  724.    Concatenation  of  Induction  Motors.    Speed  Curves. 
Z=.1—  .3/          K=.01  +  .l> 

174.  As  instance  are  given  in  Fig.  124,  the  curves  of 
total  torque,  of  torque  of  the  second  motor,  and  of  current, 
for  the  range  of  slip  from  s  =  +  1.5  to  s  =  —  .7  for  a  pair 
of  induction  motors  in  concatenation,  of  the  constants  : 

Z0  =  Z,  =  .1  -  .Bj 


As  seen,  there  are  two  ranges  of  positive  torque  for  the 
whole  system,  one  below  half  synchronism,  and  one  from 
about  two-thirds  to  full  synchronism,  and  two  ranges  of 


INDUCTION  MOTOR. 


281 


negative  torque,  or  generator  action  of  the  motor,  from  half 
to  two-third  synchronism,  and  above  full  synchronism. 

With  higher  resistance  in  the  secondary  of  the  second 
motor,  the  second  range  of  positive  torque  of  the  system 
disappears  more  or  less,  and  the  torque  curves  become  as 
shown  in  Fig.  125. 


001 

|              | 
CATENATION  jOF    IN 

SUCTION    MOTORS. 

L 

j  SPEED  CURVES 
|z=.|—  .3,j     Y4=.OI 

H-.l 

it 

rag 

RE! 

.   IN    S 

;COND 

kRY   0 

'   SECO 

NO    MC 

TOR. 

| 

H 
8000 

6000 

- 



4000 

\ 

2000 

1 

— 

— 

— 

""-s. 

\ 

I 

0 

M 

\\ 

\ 

-2000 

\\ 

X 

^ 

-4000 

£ 

/ 

f 

-60C( 

./ 

-8000 

1 

0 

9 

s 

. 

6 

j 

4 

3 

2 

j 

„ 

Fig.  125.    Concatenation  of  Induction  Motors.    Speed  Curves. 


SINGLE-PHASE  INDUCTION  MOTOR. 

175.  The  magnetic  circuit  of  the  induction  motor  at  or 
near  synchronism  consists  of  two  magnetic  fluxes  super- 
imposed upon  each  other  in  quadrature,  in  time,  and  in 
position.  In  the  polyphase  motor  these  fluxes  are  produced 
by  E.M.Fs.  displaced  in  phase.  In  the  monocyclic  motor 
one  of  the  fluxes  is  due  to  the  primary  energy  circuit,  the 
other  to  the  primary  exciting  circuit.  In  the  single-phase 


282  AL  TERN  A  TING-CURRENT  PHENOMENA. 

motor  the  one  flux  is  produced  by  the  primary  circuit,  the 
other  by  the  currents  induced  in  the  secondary  or  armature, 
which  are  carried  into  quadrature  position  by  the  rotation 
of  the  armature.  In  consequence  thereof,  while  in  all  these 
motors  the  magnetic  distribution  is  the  same  at  or  near  syn- 
chronism, and  can  be  represented  by  a  rotating  field  of 
uniform  intensity  and  uniform  velocity,  it  remains  such  in 
polyphase  and  monocyclic  motors  ;  but  in  the  single-phase 
motor,  with  increasing  slip,  —  that  is,  decreasing  speed,  — 
the  quadrature  field  decreases,  since  the  induced  armature 
currents  are  not  carried  to  complete  quadrature  position  ; 
and  thus  only  a  component  available  for  producing  the 
quadrature  flux.  Hence,  approximately,  the  quadrature  flux 
of  a  single-phase  motor  can  be  considered  as  proportional  to 
its  speed  ;  that  is,  it  is  zero  at  standstill. 

Since  the  torque  of  the  motor  is  proportional  to  the 
product  of  secondary  current  times  magnetic  flux  in  quad- 
rature, it  follows  that  the  torque  of  the  single-phase  motor 
is  equal  to  that  of  the  same  motor  under  the  same  condition 
of  operation  on  a  polyphase  circuit,  multiplied  with  the 
speed  ;  hence  equal  to  zero  at  standstill. 

Thus,  while  single-phase  induction  motors  are  quite  sat- 
isfactory at  or  near  synchronism,  their  torque  decreases 
proportionally  to  the  speed,  and  becomes  zero  at  standstill. 
That  is,  they  are  not  self-starting,  but  some  starting  device 
has  to  be  used. 

Such  a  starting  device  may  either  be  mechanical  or  elec- 
trical. All  the  electrical  starting  devices  essentially  consist 
in  impressing  upon  the  motor  at  standstill  a  magnetic  quad- 
rature flux.  This  may  be  produced  either  by  some  outside 
E.M.F.,  as  in  the  monocyclic  starting  device,  or  by  displa- 
cing the  circuits  of  two  or  more  primary  coils  from  each 
other,  either  by  mutual  induction  between  the  coils,  —  that 
is,  by  using  one  as  secondary  to  the  other,  —  or  by  impe- 
dances of  different  inductance  factors  connected  with  the 
different  primary  coils. 


INDUCTION  MOTOR.  283 

176.  The  starting-devices  of  .the  single-phase  induc- 
tion motor  by  producing  a  quadrature  magnetic  flux  can  be 
subdivided  into  three  classes  : 

1.  Phase-Splitting    Devices.      Two   or   more   primary 
circuits  are  used,  displaced  in  position  from  each  other,  and 
either  in  series  or  in  shunt  with  each  other,  or  in  any  other 
way  related,   as   by  transformation.      The  impedances  of 
these  circuits  are  made  different  from  each  other  as  much 
as  possible,  to  produce  a  phase  displacement  between  them. 
This  can  be  done  either  by  inserting  external  impedances 
into  the  circuits,  as  a  condenser  and  a  reactive  coil,  or  by 
making  the  internal  impedances  of  the  motor  circuits  differ- 
ent, as  by  making  one  coil  of  high  and  the  other  of  low 
resistance. 

2.  Inductive  Devices.     The  different  primary  circuits 
of  the  motor  are  inductively  related  to  each  other  in  such  a 
way  as  to  produce  a  phase  displacement  between  them. 
The  inductive  relation  can  be  outside  of  the  motor  or  inside, 
by  having  the  one  coil  induced  by  the  other ;  and  in  this 
latter  case  the  current  in  the  induced  coil  may  be  made 
leading,  accelerating  coil,  or  lagging,  shading  coil. 

3.  Monocyclic    Devices.     External   to   the   motor  an 
essentially  wattless  E.M.F.  is  produced  in  quadrature  with 
the  main   E.M.F.    and  impressed  upon  the  motor,  either 
directly  or   after   combination  with  the  single-phase  main 
E.M.F.     Such  wattless  quadrature  E.M.F.  can  be  produced 
by  the  common  connection  of  two  impedances  of  different 
power  factor,  as  an  inductance  and  a  resistance,  or  an  in- 
ductance and  a  condensance  connected  in  series  across  the 
mains. 

The  investigation  of  these  starting-devices  offers  a  very 
instructive  application  of  the  symbolic  method  of  investiga- 
tion of  alternating-current  phenomena,  and  a  study  thereof 
is  thus  recommended  to  the  reader.* 

»  See  paper  on  the  Single-phase  Induction  Motor,  A.I.E.E.  Transactions,  1898. 


284  ALTERNATING-CURRENT  PHENOMENA. 

177.  As  a  rule,  no  special  motors  are  built  for  single- 
phase  operation,  but  polyphase  motors  used  in  single-phase 
circuits,  since  for  starting  the  polyphase  primary  winding  is 
required,  the  single  primary  coil  motor  obviously  not  allow- 
ing the  application  of  phase-displacing  devices  for  produ- 
cing the  starting  quadrature  flux. 

Since  at  or  near  synchronism,  at  the  same  impressed 
E.M.F. — that  is,  the  same  magnetic  density  —  the  total 
voltamperes  excitation  of  the  single-phase  induction  motor 
must  be  the  same  as  of  the  same  motor  on  polyphase  circuit, 
it  follows  that  by  operating  a  quarter-phase  motor  from 
single-phase  circuit  on  one  primary  coil,  its  primary  excit- 
ing admittance  is  doubled.  Operating  a  three-phase  motor 
single-phase  on  one  circuit  its  primary  exciting  admittance 
is  trebled.  The  self-inductive  primary  impedance  is  the 
same  single-phase  as  polyphase,  but  the  secondary  impe- 
dance reduced  to  the  primary  is  lowered,  since  in  single- 
phase  operation  all  secondary  circuits  correspond  to  the 
one  primary  circuit  used.  Thus  the  secondary  impedance 
in  a  quarter-phase  motor  running  single-phase  is  reduced  to 
one-half,  in  a  three-phase  motor  running  single-phase  re- 
duced to  one-third.  In  consequence  thereof  the  slip  of 
speed  in  a  single-phase  induction  motor  is  usually  less  than 
in  a  polyphase  motor ;  but  the  exciting  current  is  consider- 
ably greater,  and  thus  the  power  factor  and  the  efficiency 
are  lower. 

The  preceding  considerations  obviously  apply  only  when 
running  so  near  synchronism  that  the  magnetic  field  of  the 
single-phase  motor  can  be  assumed  as  uniform,  that  is  the 
cross  magnetizing  flux  produced  by  the  armature  as  equal 
to  the  main  magnetic  flux. 

When  investigating  the  action  of  the  single-phase  motor 
at  lower  speeds  and  at  standstill,  the  falling  off  of  the  mag- 
netic quadrature  flux  produced  by  the  armature  current,  the 
change  of  secondary  impedance,  and  where  a  starting  device 
is  used  the  effect  of  the  magnetic  field  produced  by  the 
starting  device,  have  to  be  considered. 


INDUCTION  MOTOR.  285 

The  exciting  current  of  the  single-phase  motor  consists 
of  the  primary  exciting  current  or  current  producing  the 
main  magnetic  flux,  and  represented  by  a  constant  admit- 
tance F,,1,  the  primary  exciting  admittance  of  the  motor,  and' 
the  secondary  exciting  current,  that  is  that  component  of 
primary  current  corresponding  to  the  secondary  current 
which  gives  the  excitation  for  the  quadrature  magnetic  flux. 
This  latter  magnetic  flux  is  equal  to  the  main  magnetic  flux 
3>0  at  synchronism,  and  falls  off  with  decreasing  speed  to 
zero  at  standstill,  if  no  starting  device  is  used  or  to  4^  =  /<£0 
at  standstill  if  by  a  starting  device  a  quadrature  magnetic 
flux  is  impressed  upon  the  motor,  and  at  standstill  t  =  ratio- 
of  quadrature  or  starting  magnetic  flux  to  main  magnetic 
flux. 

Thus  the  secondary  exciting  current  can  be  represented 
by  an  admittance  Y*  which  changes  from  equality  with  the 
primary  exciting  admittance  Y^  at  synchronism,  to  Y*  =  0, 
respectively  to  Y^  —  t  Y^  at  standstill.  Assuming  thus  that 
the  starting  device  is  such  that  its  action  is  not  impaired  by 
the  change  of  speed,  at  slip  s  the  secondary  exciting  admit- 
tance can  be  represented  by  : 

Y*  =  [!-(!-/)  j]  Fo1 

The  secondary  impedance  of  the  motor  at  synchronism 
is  the  joint  impedance  of  all  the  secondary  circuits,  since  all 
secondary  circuits  correspond  to  the  same  primary  circuit, 

hence  =  -^  with  a  three-phase  secondary,  and  =  -^  with  a 

two-phase  secondary  with  impedance  Z1  per  circuit. 

At  standstill,  however,  the  secondary  circuits  correspond 
to  the  primary  circuit  only  with  their  projection  in  the  direc- 
tion of  the  primary  flux,  and  thus  as  resultant  only  one-half 
of  the  secondary  circuits  are  effective,  so  that  the  secondary 
impedance  at  standstill  is  equal  to  2  Zl  /  3  with  a  three-phase, 
and  equal  to  Z^  with  a  two-phase  secondary.  Thus  the 
effective  secondary  impedance  of  the  single-phase  motor 


286  ALTERNATING-CURRENT  PHENOMENA. 

changes  with  the  speed  and  can  at  the  slip  s  be  represented 

by  Zf  =  -  --  -^  —  -  in  a  three-phase  motor,  and  Z{  =  -  -  <p  —  -1 

in  a  two-phase  motor,  with  the  impedance  Z^  per  secondary 
circuit. 

In  the  single-phase  motor  without  starting  device,  due  to 
the  falling  off  of  the  quadrature  flux,  the  torque  at  slip  s  is  : 

T  =  a^  (I  -  s) 

In  a  single-phase  motor  with  a  starting  device  which  at 
standstill  produces  a  ratio  of  magnetic  fluxes  t,  the  torque  at 
standstill  is  ; 

TQ  =  /7I 

where   7^  =  total  torque  of   the  same  motor  on  polyphase 
circuit. 

.     Thus  denoting  the  value  —~  =  v 
&f 

the  single-phase  motor  torque  at  standstill  is  : 


and  the  single-phase  motor  torque  at  slip  s  is  : 
T  =  of  [1  -  (1  -  v)  s] 

178.  In  the  single-phase  motor  considerably  more 
advantage  is  gained  by  compensating  for  the  wattless  mag- 
netizing component  of  current  by  capacity  than  in  the 
polyphase  motor,  where  this  wattless  current  is  relatively 
small.  The  use  of  shunted  capacity,  however,  has  the  dis- 
advantage of  requiring  a  wave  of  impressed  E.M.F.  very 
close  to  sine  shape  ;  since  even  with  a  moderate  variation 
from  sine  shape  the  wattless  charging  current  of  the  con- 
denser of  higher  frequency  may  lower  the  power  factor 
more  than  the  compensation  for  the  wattless  component  of 
the  fundamental  wave  raises  it,  as  will  be  seen  in  the  chap- 
ter on  General  Alternating  Current  Waves. 

Thus  the  most  satisfactory  application  of  the  condenser 
in  the  single-phase  motor  is  not  in  shunt  to  the  primary 


INDUCTION  MOTOR.  287 

circuit,  but  in  a  tertiary  circuit ;  that  is,  in  a  circuit  stationary 
with  regard  to  the  primary  impressed  circuit,  but  induced 
by  the  revolving  secondary  circuit. 

In  this  case  the  condenser  is  supplied  with  an  E.M.F. 
transformed  twice,  from  primary  to  secondary,  and  from 
secondary  to  tertiary,  through  multitooth  structures  in  a 
uniformly  revolving  field,  and  thus  a  very  close  approxi- 
mation to  sine  wave  produced  at  the  condenser,  irrespective 
of  the  wave  shape  of  primary  impressed  E.M.F. 

With  the  condenser  connected  into  a  tertiary  circuit  of 
a  single-phase  induction  motor,  the  wattless  magnetizing 
current  of  the  motor  is  supplied  by  the  condenser  in  a 
separate  circuit,  and  the  primary  coil  carries  the  energy  cur- 
rent only,  and  thus  the  efficiency  of  the  motor  is  essentially 
increased. 

The  tertiary  circuit  may  be  at  right  angles  to  the  pri- 
mary, or  under  any  other  angle.  Usually  it  is  applied  on  an 
angle  of  60°,  so  as  to  secure  a  mutual  induction  between 
tertiary  and  primary  for  starting,  which  produces  in  start- 
ing in  the  condenser  a  leading  current,  and  gives  the  quad- 
rature magnetic  flux  required. 

179.  The  most  convenient  way  to  secure  this  arrange- 
ment is  the  use  of  a  three-phase  motor  which  with  two  of 
its  terminals  1-2,  is  connected  to  the  single-phase  mains, 
and  with  terminals  1  and  3  to  a  condenser. 

Let  YQ  =  g0  -\-jb0  =  primary  exciting  admittance  of  the 
motor  per  delta  circuit. 

Z0  =  r0  —  jxQ  =  primary  self-inductive  impedance  per 
delta  circuit. 

Z^  =  i\  —jx^  =  secondary  self-inductive  impedance  per 
delta  circuit  reduced  to  primary. 

Let 

Ys  =  gs  —  jb9  =  admittance  of   the  condenser  connected 
between  terminals  1  and  3. 


288  ALTERNATING-CURRENT  PHENOMENA. 

If  then,  as  single-phase  motor, 

/  =  ratio  of   auxiliary  quadrature  flux  to  main  flux  in 
starting, 

h  =  ratio   of    E.M.F.   induced   in  condenser  circuit  to 

E.M.F.  induced  in  main  circuit  in  starting, 
starting  torque 


It  is  single-phase 

Fo1  =  1.5  Y0  =  1.5  (£•„  +/£0)  =  primary   exciting    admit- 

tance, 
Y?  =  1.5  Y0  [1  -  (1  -  0  s] 

=  1.5  (g0  +/£<))  [1  —  (1  —  0  J]  =  secondary  exciting 

admittance  at  slip  s. 

Z0l  =  ?^°  =  2fo~^*o)  =  primary    self-inductive    impe- 
o  o 

dance. 

Zxi  =  £L±^  Zi  =  ^L  +  ^  (ri  -jsxj  =  secondary    self- 
o  o 

inductive  impedance. 

Z,1  =  ^  =  2  (r°  ~  ***>  =  tertiary    self-inductive    impe- 
o  o 

dance  of  motor. 
Thus, 

Y4  =  -^r  -  T-  =  total  admittance  of  tertiary  circuit. 


Since  the  E.M.F.  induced  in  the  tertiary  circuit  decreases 
from  e  at  synchronism  to  he  at  standstill,  the  effective  ter- 
tiary admittance  or  admittance  reduced  to  an  induced  E.M.F. 
e  is  at  slip  s 

Y?  =  [!-(!-*)  s]  Y4 
Let  then, 

e  =  counter  E.M.F.  of  primary  circuit, 
s  =  slip. 


INDUCTION  MOTOR.  289 


We  have, 
secondary  load  current 

3se 


(1  +  s)  (r,  -jsx,) 
secondary  exciting  current 

secondary  condenser  current 
thus,  total  secondary  current 
primary  exciting  current 


thus,  total  primary  current 

/o  =  71  +  /o1 
=  /,  +  /,  + 

=  '  (*i  +  A) 
primary  impressed  E.M.F. 


thus,  main  counter  E.M.F. 


or, 


and,  absolute 


V^2  +  c* 
hence,  primary  current 


T_slW    +    % 

J*  -  e°  v  f*  +  ^ 


290  ALTERNATING-CURRENT  PHENOMENA. 

voltampere  input, 

Qo  =  **!» 
power  input 


*t  —         Oo       —     O  2     ,          2 

6j     T  '2 

torque  at  slip  .$• 

2^=  r1  [i  -  (i  -  v)  s] 


and,  power  output 


and  herefrom  in  the  usual  manner  the  efficiency,  apparent 
efficiency,  torque  efficiency,  apparent  torque  efficiency,  and 
power  factor. 

The  derivation  o.*  the  constants  /,  //,  v,  which  have  to  be 
determined  before  calculating  the  motor,  is  as  follows  : 

Let    <?0  =  single-phase  impressed  E.M.F., 

Y  —  total  stationary  admittance  of  motor  per  delta  cir- 

cuit, 
Ez  =  E.M.F.  at  condenser  terminals  in  starting. 

In  the  circuit  between  the  single-phase  mains  from  ter- 
minal 1  over  terminal  3  to  2,  the  admittances  Y  +  Y8,  and  Y, 
are  connected  in  series,  and  have  the  respective  E.M.Fs.  E^ 
and  e0  -  Ey  It  is  thus, 

Y+  Ys+  Y=e0-£t+£s, 

since  with  the  same  current  passing  through  both  circuits, 
the  impressed  E.M.Fs.  are  inverse  proportional  to  the  re- 
spective admittances. 

Thus, 


INDUCTION  MOTOR.  291 

and  quadrature  E.M.F. 

hence 
thus 


Since  in  the  three-phase  E.M.F.  triangle,  the  altitude 
corresponding  to  the  quadrature  magnetic  flux  =  — y= ,  and 

the  quadrature  and  main  fluxes  are  equal,  in  the  single-phase 
motor  the  ratio  of  quadrature  to  main  flux  is 

/  =  — 2  =  1.155  Aa 

V3 

From  /,  v  is  derived  as  shown  in  the  preceding. 

For  further  discussion  on  the  Theory  and  Calculation  of 
the  Single-phase  Induction  Motor,  see  American  Institute 
Electrical  Engineers  Transactions,  January,  1900. 


SYNCHRONOUS  INDUCTION  MOTOR. 

180.  The  induction  motor  discussed  in  the  foregoing 
consists  of  one  or  a  number  of  primary  circuits  acting  upon 
a  movable  armature  which  comprises  a  number  of  closed 
secondary  circuits  displaced  from  each  other  in  space  so  as 
to  offer  a  resultant  circuit  in  any  direction.  In  consequence 
thereof  the  motor  can  be  considered  as  a  transformer,  having 
to  each  primary  circuit  a  corresponding  secondary  circuit, 
—  a  secondary  coil,  moving  out  of  the  field  of  the  primary 
coil,  being  replaced  by  another  secondary  coil  moving  into 
the  field. 

In  such  a  motor  the  torque  is  zero  at  synchronism,  posi- 
tive below,  and  negative  above,  synchronism. 

If,  however,  the  movable  armature  contains  one  closed 
circuit  only,  it  offers  a  closed  secondary  circuit  only  in  the 
direction  of  the  axis  of  the  armature  coil,  but  no  secondary 
circuit  at  right  angles  therewith.  That  is,  with  the  rotati  .n 


292  ALTERNATING-CURRENT  PHENOMENA. 

of  the  armature  the  secondary  circuit,  corresponding  to  a 
primary  circuit,  varies  from  short  circuit  at  coincidence  of 
the  axis  of  the  armature  coil  with  the  axis  of  the  primary 
coil,  to  open  circuit  in  quadrature  therewith,  with  the 
periodicity  of  the  armature  speed.  That  is,  the  apparent 
admittance  of  the  primary  circuit  varies  periodically  from 
open-circuit  admittance  to  the  short-circuited  transformer 
admittance. 

At  synchronism  such  a  motor  represents  an  electric  cir- 
cuit of  an  admittance  varying  with  twice  the  periodicity  of 
the  primary  frequency,  since  twice  per  period  the  axis  of  the 
armature  coil  and  that  of  the  primary  coil  coincide.  A  vary- 
ing admittance  is  obviously  identical  in  effect  with  a  varying 
reluctance,  which  will  be  discussed  in  the  chapter  on  reac- 
tion machines.  That  is,  the  induction  motor  with  one 
•closed  armature  circuit  is,  at  synchronism,  nothing  but  a 
reaction  machine,  and  consequently  gives  zero  torque  at 
synchronism  if  the  maxima  and  minima  of  the  periodically 
varying  admittance  coincide  with  the  maximum  and  zero 
values  of  the  primary  circuit,  but  gives  a  definite  torque  if 
they  are  displaced  therefrom.  This  torque  may  be  positive 
or  negative  according  to  the  phase  displacement  between 
admittance  and  primary  circuit  ;  that  is,  the  lag  or  lead 
of  the  maximum  admittance  with  regard  to  the  primary 
maximum.  Hence  an  induction  motor  with  single-armature 
circuit  at  synchronism  acts  either  as  motor  or  as  alternat- 
ing-current generator  according  to  the  relative  position  of 
the  armature  circuit  to  the  primary  circuit.  Thus  it  can  be 
called  a  synchronous  induction  motor  or  synchronous  in- 
duction generator,  since  it  is  an  induction  machine  giving 
torque  at  synchronism. 

Power  factor  and  apparent  efficiency  of  the  synchron- 
ous induction  motor  as  reaction'  machine  are  very  low. 
Hence  it  is  of  practical  application  only  in  cases  where  a 
small  amount  of  power  is  required  at  synchronous  rotation, 
and  continuous  current  for  field  excitation  is  not  available. 


INDUCTION  MOTOR.  293 

The  current  induced  in  the  armature  of  the  synchronous 
induction  motor  is  of  double  the  frequency  impressed  upon 
the  primary. 

Below  and  above  synchronism  the  ordinary  induction 
motor,  or  induction  generator,  torque  is  superimposed  upon 
the  synchronous  induction  machine  torque.  Since  with  the 
frequency  of  slip  the  relative  position  of  primary  and  of 
secondary  coil  changes,  the  synchronous  induction  machine 
torque  alternates  periodically  with  the  frequency  of  slip. 
That  is,  upon  the  constant  positive  or  negative  torque  be- 
low or  above  synchronism  an  alternating  torque  of  the  fre- 
quency of  slip  is  superimposed,  and  thus  the  resultant 
torque  pulsating  with  a  positive  mean  value  below,  a  nega- 
tive mean  value  above,  synchronism. 

When  started  from  rest,  a  synchronous  induction  motor 
will  accelerate  like  an  ordinary  single-phase  induction  mo- 
tor, but  not  only  approach  synchronism,  as  the  latter  does, 
but  run  up  to  complete  synchronism  under  load.  When 
approaching  synchronism  it  makes  definite  beats  with  the 
frequency  of  slip,  which  disappear  when  synchronism  is 
reached. 

THE  HYSTERESIS  MOTOR. 

181.  In  a  revolving  magnetic  field,  a  circular  iron  disk, 
or  iron  cylinder  of  uniform  magnetic  reluctance  in  the 
direction  of  the  revolving  field,  is  set  in  rotation,  even  if 
subdivided  so  as  to  preclude  the  induction  of  eddy  currents. 
This  rotation  is  due  to  the  effect  of  hysteresis  of  the  revolv- 
ing disks  or  cyclinder,  and  such  a  motor  may  thus  be  called 
a  hysteresis  motor. 

Let  /  be  the  iron  disk  exposed  to  a  rotating  magnetic 
field  or  resultant  M.M.F.  The  axis  of  resultant  magneti- 
zation in  the  disk  /  does  not  coincide  with  -the  axis  of  the 
rotating  field,  but  lags  behind  the-  latter,  thus  producing  a 
couple.  That  is,  the  component  of  magnetism  in  a  direction 
of  the  rotating  disk,  /,  ahead  of  the  axis  of  rotating  M.M.F., 
is  rising,  thus  below,  and  in  a  direction  behind  the  axis 


294  AL  TERN  A  TING-CURRENT  PHENOMENA. 

of  rotating  M.M.F.  decreasing;  that  is,  above  proportion- 
ality with  the  M.M.F.,  in  consequence  of  the  lag  of  magnet- 
ism in  the  hysteresis  loop,  and  thus  the  axis  of  resultant 
magnetism  in  the  iron  disk,  /,  does  not  coincide  with  the 
axis  of  rotating  M.M.F.,  but  is  shifted  backwards  by  an 
angle,  a,  which  is  the  angle  of  hysteretic  lead  in  Chapter 
X.,  §  79. 

The  induced  magnetism  gives  with  the  resultant  M.M.F. 
a  mechanical  couple,  — 


T=  mF&  sin  a, 

where 

F=  resultant  M.M.F., 

<£  =  resultant  magnetism, 

a  =  angle  of  hysteretic  advance  of  phase, 

m  =  a  constant. 

The  apparent  or  voltampere  input  of  the  motor  is,  — 
Q  =  mF®. 

Thus  the  apparent  torque  efficiency,  — 

T 

2  =  sma, 

and  the  power  of  the  motor  is,  — 

P  =  (1  —  s)  T=  (1  —  s)  m  F<$>  sin  a, 
where 

s  =  slip  as  fraction  of  synchronism. 

The  apparent  efficiency  is,  — 

P 

-  =  (!_*)  sin  a. 

Since  in  a  magnetic  circuit  containing  an  air  gap  the 
angle  a  is  extremely  small,  a-  few  degrees  only,  it  follows 
that  the  apparent  efficiency  of  the  hysteresis  motor  is  ex- 
tremely low,  the  motor  consequently  unsuitable  for  produ- 
cing larger  amounts  of  mechanical  work. 


INDUCTION  MOTOR.  295 

From  the  equation  of  torque  it  follows,  however,  that  at 
constant  impressed  E.M.F.,  or  current,  —  that  inconstant 
F,  —  the  torque  is  constant  and  independent  of  the  speed  ; 
and  therefore  such  a  motor  arrangement  is  suitable,  and 
occasionally  used  as  alternating-current  meter. 

The  same  result  can  be  reached  from  a  different  point 
of  view.  In  such  a  magnetic  system,  comprising  a  mov- 
able iron  disk,  /,  of  uniform  magnetic  reluctance  in  a 
revolving  field,  the  magnetic  reluctance  —  and  thus  the  dis- 
tribution of  magnetism  —  is  obviously  independent  of  the 
speed,  and  consequently  the  current  and  energy  expenditure 
of  the  impressed  M.M.F.  independent  of  the  speed  also.  If, 
now,  — 

V  '=  volume  of  iron  of  the  movable  part, 
B  =  magnetic  density, 
and  77  =  coefficient  of  hysteresis, 

the  energy  expended  by  hysteresis  in  the  movable  disk,  /,  is 
per  cycle,  — 

IV,  =  V^B™, 

hence,  if  N=  frequency,  the  energy  supplied  by  the  M.M.F. 
to   the    rotating    iron    disk    in  the  hysteretic   loop  of   the 

M.M.F.  is,  — 

P  = 


At  the  slip,  s  N,  that  is,  the  speed  (1  —  s)  N,  the  energy 
xpended  by  hysteresis  in  the  rotating  disk  is,  however,  — 


Hence,  in  the  transfer  from  the  stationary  to  the  revolv- 
ing member  the  magnetic  energy,  — 


has  disappeared,  and   thus   reappears   as  mechanical  work, 
and  the  torque  is,  — 

'-p^iprW' 

that  is,  independent  of  the  speed. 


296  AL  TERNA  TING-CURRENT  PHENOMENA. 

Since,  as  seen  in  Chapter  X.,  sin  a  is  the  ratio  of  the 
energy  of  the  hysteretic  loop  to  the  total  apparent  energy, 
in  voltampere,  of  the  magnetic  cycle,  it  follows  that  the 
apparent  efficiency  of  such  a  motor  can  never  exceed  the 
value  (1  —  s)  sin  a,  or  a  fraction  of  the  primary  hysteretic 
energy. 

The  primary  hysteretic  energy  of  an  induction  motor,  as 
represented  by  its  conductance,  g,  being  a  part  of  the  loss 
in  the  motor,  and  thus  a  very  small  part  of  its  output  only, 
it  follows  that  the  output  of  a  hysteresis  motor  is  a  very 
small  fraction  only  of  the  output  which  the  same  magnetic 
structure  could  give  with  secondary  short-circuited  winding, 
as  regular  induction  motor. 

As  secondary  effect,  however,  the  rotary  effort  of  the 
magnetic  structure  as  hysteresis  motor  appears  more  or  less 
in  all  induction  motors,  although  usually  it  is  so  small  as  to 
be  neglected. 

If  in  the  hysteresis  motor  the  rotary  iron  structure  has 
not  uniform  reluctance  in  all  directions  —  but  is,  for  in- 
stance, bar-shaped  or  shuttle-shaped  —  on  the  hysteresis 
motor  effect  is  superimposed  the  effect  of  varying  magnetic 
reluctance,  which  tends  to  accelerate  the  motor  to  syn- 
chronism, and  maintain  it  therein,  as  shall  be  more  fully 
investigated  under  "  Reaction  Machine  "  in  Chapter  XX. 


ALTERNATING-CURRENT  GENERATOR.  297 


CHAPTER    XVII. 

ALTERNATING-CURRENT    GENERATOR. 

182.  In  the  alternating-current  generator,  E.M.F.  is 
induced  in  the  armature  conductors  by  their  relative  motion 
through  a  constant  or  approximately  constant  magnetic 
field. 

When  yielding  current,  two  distinctly  different  M.M.Fs. 
are  acting  upon  the  alternator  armature  —  the  M.M.F.  of 
the  field  due  to  the  field-exciting  'spools,  and  the  M.M.F. 
of  the  armature  current.  The  former  is  constant,  or  approx- 
imately so,  while  the  latter  is  alternating,  and  in  synchro- 
nous motion  relatively  to  the  former  ;  hence,  fixed  in  space 
relative  to  the  field  M.M.F.,  or  uni-directional,  but  pulsating 
in  a  single-phase  alternator.  In  the  polyphase  alternator, 
when  evenly  loaded  or  balanced,  the  resultant  M.M.F.  of 
the  armature  current  is  more  or  less  constant. 

The  E.M.F.  induced  in  the  armature  is  due  to  the  mag- 
netic flux  passing  through  and  interlinked  with  the  arma- 
ture conductors.  This  flux  is  produced  by  the  resultant  of 
both  M.M.Fs.,  that  of  the  field,  and  that  of  the  armature. 

On  open  circuit,  the  M.M.F.  of  the  armature  is  zero,  and 
the  E.M.F.  of  the  armature  is  due  to  the  M.M.F.  of  the 
field  coils  only.  In  this  case  the  E.M.F.  is,  in  general,  a 
maximum  at  the  moment  when  the  armature  coil  faces  the 
position  midway  between  adjacent  field  coils,  as  shown  in 
Fig.  126,  and  thus  incloses  no  magnetism.  The  E.M.F. 
wave  in  this  case  is,  in  general,  symmetrical. 

An  exception  from  this  statement  may  take  place  only 
in  those  types  of  alternators  where  the  magnetic  reluctance 
of  the  armature  is  different  in  different  directions  ;  thereby, 


298  AL  TERNA  TING-CURRENT  PHENOMENA. 

during  the  synchronous  rotation  of  the  armature,  a  pulsa- 
tion of  the  magnetic  flux  passing  through  it  is  produced. 
This  pulsation  of  the  magnetic  flux  induces  E.M.F.  in  the 
field  spools,  and  thereby  makes  the  field  current  pulsating 
also.  Thus,  we  havet  in  this  case,  even  on  open  circuit,  no 


Fig.  126. 

rotation  through  a  constant  magnetic  field,  but  rotation 
through  a  pulsating  field,  which  makes  the  E.M.F.  wave 
unsymmetrical,  and  shifts  the  maximum  point  from  its  the- 
oretical position  midway  between  the  field  poles.  In  gen- 
eral this  secondary  reaction  can  be  neglected,  and  the  field 
M.M.F.  be  assumed  as  constant. 

The  relative  position  of  the  armature  M.M.F.  with  re- 
spect to  the  field  M.M.F.  depends  upon  the  phase  rela- 
tion existing  in  the  electric  circuit.  Thus,  if  there  is  no 
displacement  of  phase  between  current  and  E.M.F.,  the 
current  reaches  its  maximum  at  the  same  moment  as  the 
E.M.F. ;  or,  in  the  position  of  the  armature  shown  in  Fig. 
126,  midway  between  the  field  poles.  In  this  case  the  arma- 
ture current  tends  neither  to  magnetize  nor  demagnetize  the 
field,  but  merely  distorts  it  ;  that  is,  demagnetizes  the  trail- 
ing-pole  corner,  a,  and  magnetizes  the  leading-pole  corner, 
b.  A  change  of  the  total  flux,  and  thereby  of  the  resultant 
E.M.F.,  will  take  place  in  this  case  only  when  the  magnetic 
densities  are  so  near  to  saturation  that  the  rise  of  density 
at  the  leading-pole  corner  will  be  less  than  the  decrease  of 


AL  TERN  A  TING-CURRENT  GENERA  TOR. 


299 


density  at  the  trailing-pole  corner.  Since  the  internal  self- 
inductance  of  the  alternator  itself  causes  a  certain  lag  of 
the  current  behind  the  induced  E.M.F.,  this  condition  of  no 
displacement  can  exist  only  in  a  circuit  with  external  nega- 
tive reactance,  as  capacity,  etc. 

If  the  armature  current  lags,  it  reaches  the  maximum 
later  than  the  E.M.F.  ;  that  is,  in  a  position  where  the 
armature  coil  partly  faces  the  following-field  pole,  as  shown 
in  diagram  in  Fig.  127.  Since  the  armature  current  flows 


Fig.  127. 


in  opposite  direction  to  the  current  in  the  following-field 
pole  (in  a  generator),  the  armature  in  this  case  will  tend  to 
demagnetize  the  field. 

If,  however,  the  armature  current  leads, — that  is,  reaches 
its  maximum  while  the  armature  coil  still  partly  faces  the 


Fig.  128. 


preceding-field  pole,  as  shown  in  diagram  Fig.  128, — it  tends 
to  magnetize  this  field  coil,  since  the  armature  current  flows 
in  the  same  direction  with  the  exciting  current  of  the  pre- 
ceding-field spools. 


300  ALTERNA TING-CURRENT  PHENOMENA. 

Thus,  with  a  leading  current,  the  armature  reaction  of 
the  alternator  strengthens  the  field,  and  thereby,  at  con- 
stant-field excitation,  increases  the  voltage  ;  with  lagging 
current  it  weakens  the  field,  and  thereby  decreases  the  vol- 
tage in  a  generator.  Obviously,  the  opposite  holds  for  a 
synchronous  motor,  in  which  the  armature  current  flows  in 
the  opposite  direction  ;  and  thus  a  lagging  current  tends  to 
magnetize,  a  leading  current  to  demagnetize,  the  field. 

183.  The  E.M.F.  induced  in  the  armature  by  the  re- 
sultant magnetic  flux,  produced  by  the  resultant  M.M.F.  of 
the  field  and  of  the  armature,  is  not  the  terminal  voltage 
of  the  machine ;  the  terminal  voltage  is  the  resultant  of  this 
induced  E.M.F.  and  the  E.M.F.  of  self-inductance  and  the 
E.M.F.  representing  the  energy  loss  by  resistance  in  the 
alternator  armature.  That  is,  in  other  words,  the  armature 
current  not  only  opposes  or  assists  the  field  M.M.F.  in  cre- 
ating the  resultant  magnetic  flux,  but  sends  a  second  mag- 
netic flux  in  a  local  circuit  through  the  armature,  which 
flux  does  not  pass  through  the  field  spools,  and  is  called  the 
magnetic  flux  of  armature  self-inductance. 

Thus  we  have  to  distinguish  in  an  alternator  between 
armature  reaction,  or  the  magnetizing  action  of  the  arma- 
ture upon  the  field,  and  armature  self-inductance,  or  the 
E.M.F.  induced  in  the  armature  conductors  by  the  current 
flowing  therein.  This  E.M.F.  of  self-inductance  is  (if  the 
magnetic  reluctance,  and  consequently  the  reactance,  of 
the  armature  circuit  is  assumed  as  constant)  in  quadrature 
behind  the  armature  current,  and  will  thus  combine  with 
the  induced  E.M.F.  in  the  proper  phase  relation.  Obvi- 
ously the  E.M.F.  of  self-inductance  and  the  induced  E.M.F. 
do  not  in  reality  combine,  but  their  respective  magnetic 
fluxes  combine  in  the  armature  core,  where  they  pass  through 
the  same  structure.  These  component  E.M.Fs.  are  there- 
fore mathematical  fictions,  but  their  resultant  is  real.  This 
means  that,  if  the  armature  current  lags,  the  E.M.F.  of  self- 


ALTERNATING-CURRENT  GENERATOR.  301 

inductance  will  be  more  than  90°  behind  the  induced  E.M.F., 
and  therefore  in  partial  opposition,  and  will  tend  to  reduce 
the  terminal  voltage.  On  the  other  hand,  if  the  armature 
current  leads,  the  E.M.F.  of  self-inductance  will  be  less 
than  90°  behind  the  induced  E.M.F.,  or  in  partial  conjunc- 
tion therewith,  and  increase  the  terminal  voltage.  This 
means  that  the  E.M.F.  of  self -inductance  increases  the  ter- 
minal voltage  with  a  leading,  and  decreases  it  with  a  lagging 
current,  or,  in  other  words,  acts  in  the  same  manner  as  the 
armature  reaction.  For  this  reason  both  actions  can  be 
combined  in  one,  and  represented  by  what  is  called  the  syn- 
cJironous  reactance  of  the  alternator.  In  the  following,  we 
shall  represent  the  total  reaction  of  the  armature  of  the 
alternator  by  the  one  term,  synchronous  reactance.  While 
this  is  not  exact,  as  stated  above,  since  the  reactance  should 
be  resolved  into  the  magnetic  reaction  due  to  the  magnet- 
izing action  of  the  armature  current,  and  the  electric  reac- 
tion due  to  the  self-induction  of  the  armature  current,  it  is 
in  general  sufficiently  near  for  practical  purposes,  and  well 
suited  to  explain  the  phenomena  taking  place  under  the 
various  conditions  of  load.  This  synchronous  reactance,  x, 
Is  frequently  not  constant,  but  is  pulsating,  owing  to  the 
synchronously  varying  reluctance  of  the  armature  magnetic 
circuit,  and  the  field  magnetic  circuit ;  it  may,  however,  be 
considered  in  what  follows  as  constant ;  that  is,  the  E.M.Fs. 
induced  thereby  may  be  represented  by  their  equivalent  sine 
waves.  A  specific  discussion  of  the  distortions  of  the  wave 
shape  due  to  the  pulsation  of  the  synchronous  reactance  is 
found  in  Chapter  XX.  The  synchronous  reactance,  x,  is 
not  a  true  reactance  in  the  ordinary  sense  of  the  word,  but 
an  equivalent  or  effective  reactance.  Sometimes  the  total 
effects  taking  place  in  the  alternator  armature,  are  repre- 
sented by  a  magnetic  reaction,  neglecting  the  self -inductance.' 
altogether,  or  rather  replacing  it  by  an  increase  of  the  arma- 
ture reaction  or  armature  M.M.F.  to  such  a  value  as  to  in- 
clude the  self-inductance.  This  assumption  is  mostly  made 
in  the  preliminary  designs  of  alternators. 


"302  ALTERNATING-CURRENT  PHENOMENA. 

184.  Let  E0  =  induced  E.M.F.  of  the  alternator,  or  the 
E.M.F.  induced  in  the  armature  coils  by  their  rotation 
through  the  constant  magnetic  field  produced  by  the  cur- 
rent in  the  field  spools,  or  the  open  circuit  voltage,  more 
properly  called  the  "nominal  induced  E.M.F.,"  since  in 
reality  it  does  not  exist,  as  before  stated. 


Then  E0 

where 

n    =  total  number  of  turns  in  series  on  the  armature, 

JV  =  frequency, 

M  =  total  magnetic  flux  per  field  pole. 

Let  x0  =  synchronous  reactance, 

r0  =  internal  resistance  of  alternator  ; 
then  Z0  —  r0  —  j  x0  =  internal  impedance. 

If  the  circuit  of  the  alternator  is  closed  by  the  external 
impedance, 

Z  =  r-jx, 
the  current  is 

E0  E0 


or,  /= 

and,  terminal  voltage, 


or, 


+x- 


ALTERNA  TING-CURRENT  GENERA  TOR. 


303 


or,  expanded  in  a  series, 


As  shown,  the  terminal  voltage  varies  with  the  condi- 
tions of  the  external  circuit. 

185.    As  an   instance,   in   Figs.   129-134,   at    constant 
induced  E.M.F., 

Eo  =  2500  ; 


.  ^ 

/ 

' 

x\ 

\ 

*-     — 



/ 

/ 

\ 
\ 
\ 

\ 

\ 

/ 

\ 
i 

/ 

\ 

***>. 

1 

/ 

/ 

^ 

X^o 

I 
1 

i 

^J 

\ 

4S    . 

( 

.1 

'/ 

\ 

\ 
\ 

Si 
&' 

> 

\ 

\ 

n 

2°' 

^ 

f 

\ 

I 

\ 

1 

/ 

F 

ELD 

CHA 

MCI 

ERIS 

TIC 

\ 

1 
1 

1 

E0= 

1 

250( 
R  = 

>,  Zo-MOj, 
E,  xko 

\ 

I 
,    1 

1 
1 
1 

\ 

1 

± 

20        10         60         80        100       180       140       160       18P       2 

X)       2 

0       210        2 

0 

Fig.  129.    Field  Characteristic  of  Alternator  on  Non-inductive  Load. 

'  + 

and  the  values  of  the  internal  impedance, 

z0  =  r0  -jXo  =  i  -  ioy. 

With  the  current  /  as  abscissae,  the  terminal  voltages  E 
as  ordinates  in  drawn  line,  and  the  kilowatts  output,  =  /2  r, 
in  dotted  lines,  the  kilovolt-amperes  output,  =  /  £,  in  dash- 


304 


AL  TEKNA  TING-CURRENT  PHENOMENA. 


dotted  lines,  we  have,  for  the  following  conditions  of  external 

circuit  : 

In  Fig.  129,  non-inductive  external  circuit,  x  =  0. 

In  Fig.  130,  inductive  external  circuit,  of  the  condition,  r  /  x 
=  -f  .75,  with  a  power  factor,  .6. 

In  Fig.  131,  inductive  external  circuit,  of  the  condition,  r=  <>, 
with  a  power  factor,  0. 

In  Fig.  132,  external  circuit  with  leading  current,  of  the  condi- 
tion, r/x  =  —  .75,  with  a  power  factor,  .6. 

In  Fig.  133,  external  circuit  with  leading  current,  of  the  condi- 
tion, r  =  0,  with  a  power  factor,  0. 

In  Fig.  134,  all  the  volt-ampere  curves  are  shown  together  as 
complete  ellipses,  giving  also  the  negative  or  synchronous 
motor  part  of  the  curves. 


\ 

E72 

FIE 
500, 

.D  CHARA 
Zf  MOj.  i 

CTERIST(C 

-.75jop60^P.F 

"\ 

\ 

S 

\ 

\ 

\ 

-^  

^X 

\ 

*\ 

I* 

/ 

S 

fe 

\ 

II* 

So 

>/ 

X 

\ 

^ 

"i 

x'' 

\ 

\ 

/ 

J 

^ 

\ 

\ 

/ 

X^N 

\ 

/ 

^ 

\\ 

\v 

(/_ 

\ 

^ 

20        40        60        80       1 

K»       120        140       1 

H)       180       200       220      glQ      20 

0  Amp 

Fig.  130.    Field  Characteristic  of  Alternator,  at  60%  Power-factor  on  Inductive  Load. 

Such  a  curve  is  called  a  field  characteristic. 

As  shown,  the  E.M.F.  curve  at  non-inductive  load  is 
nearly  horizontal  at  open  circuit,  nearly  vertical  at  short 
circuit,  and  is  similar  to  an  arc  of  an  ellipse. 


ALTERNATING-CURRENT  GENERATOR.  305 


\ 

s, 

FIELD  CHARACTt 
:0=25OO,  Z?1-10j,  r  = 

RISTIC 

o,  90°  Lag 

\ 

\ 

1  R  = 

0. 

\ 

\ 

\ 

\ 

\ 

\ 

k 

o  » 

>C" 

-X 

A 

/ 

S 

\%< 

\ 

o  2" 

X     X 

t 

s 

% 

\ 

/ 

/ 

\ 

\ 

\ 

/ 

\ 

\ 

> 

/ 

\ 

\ 

\ 

/ 

\ 

0 

/ 

s, 

\ 

Fig.  131.    Field  Characteristic  of  Alternator,  on  Wattless  Inductive  Load. 


5 
I 

li'.'U 

1000 

HM 

^ 

^ 

Ns 

V 

x^ 

\ 

.'.X'OU 

^ 

X" 

\ 

X 

? 

X 

F 

EU 

Ch 

AR 

ACT 

ER 

ST 

c 

E 

f  2 

50C 

),  Z 

1-1 

3j.  : 

=  -.75  c 

r  6 

3^F 

.F. 

iloo 

/ 

«••"" 

fc    y 

^ 

KM 

£ 

/ 

/ 

/ 

f 

ItilK 

< 

/ 

/ 

j 

/ 

,-* 

'"' 

/ 

j 

^ 

400 

" 

lain.. 

^ 

s 

v£ 

-- 

1 

> 

> 

, 

,.»*' 

/ 

/ 

j 

800 

f* 

. 

X 

/ 

/ 

/ 

, 

7 

,,*" 

/ 

/ 

/ 

m 

/- 

-*''" 

A 

-n  pe 

•M 

/y 

/x. 

**' 

;-r 

*•"' 

1 

B 

, 

£ 

I 

| 

2 

0^ 

**•!•• 

0 

• 

0 

m 

Fig.   732.     Field  Characteristic  of  Alternator,  at  60%  Power-factor  on  Condenser  Load. 


306 


AL  TERNA  TING-CURRENT  PHENOMENA. 


1  I  1  1 

'/ 

FIE 

LD  CHARACTERISTIC 

/ 
/ 

i 
/ 

f 

E0-2500,  Zo-1-IOj, 
=  o.  90°Leading  Current 

/ 

/ 

I'R 

=  O 

L 

/ 

/ 

/ 

/ 
/ 

7 

/ 

/ 

r     tu 

/ 

/ 

2 

/ 

1 

/ 

? 

/ 

/ 

/ 

s 

/ 

?/ 

r 

/ 

J 

/ 

^ 

*X 

/ 

/ 

7 

I* 

11 

^ 

/ 

/ 

^x 

/ 

// 

/ 

// 

/ 

/ 

// 

! 

/ 

/ 

/ 

I/ 

/ 

/ 

// 

/ 

/ 

/    / 

/ 

/ 

g 

/ 

^-x 

^ 

x'' 

xlO 

3-  A, 

nps. 

fig.  133.    Field  Characteristic  of  Alternator,  on  Wattless  Condenser  Load. 

With  reactive  load  the  curves  are  more  nearly  straight 
lines. 

The  voltage  drops  on  inductive,  rises  on  capacity  load. 

The  output  increases  from  zero  at  open  circuit  to  a  maxi- 
mum, and  then  decreases  again  to  zero  at  short  circuit. 


AL  TERN  A  TING-CURRENT  GENERA  TOR. 


307 


M 


VK 


4^z 


W 


Fig.  134.    Field  Characteristic  of  Alternator. 

186.  The  dependence  of  the  terminal  voltage,  E,  upon 
the  phase  relation  of  the  external  circuit  is  shown  in  Fig. 
135,  which  gives,  at  impressed  E.M.F., 

E0  =  2,500  volts, 
for  the  currents, 

1=  50,  100,  150,  200,  250  amperes, 

the  terminal  voltages,  E,  as  ordinates,  with  the  inductance 
factor  of  the  external  circuit, 


as  abscissas. 

187.    If  the  internal  impedance  is  negligible  compared 
with  the  external  impedance,  then,  approximately, 


w 


308 


AL  TERNA  TING-CURRENT  PHENOMENA, 


'      .C      .5      .4      .3      .2      .1      0      -.1    -.2     -.3    -.1     -.5     -.0     -.7    -.8 

Fig.  135.    Regulation  of  Alternator  on  Various  Loads. 

that  is,  an  alternator  with  small  internal  resistance  and  syn- 
chronous reactance  tends  to  regulate  for  constant  terminal 
voltage. 

Every  alternator  does  this  near  open  circuit,  especially 
on  non-inductive  load. 

Even  if  the  synchronous  reactance,  x0 ,  is  not  quite  neg- 
ligible, this  regulation  takes  place,  to  a  certain  extent,  on 
non-inductive  circuit,  since  for 


*  =  0,   E 


and  thus  the  expression  of  the  terminal  voltage,  E,  contains 
the  synchronous  reactance,  x0,  only  as  a  term  of  second 
order  in  the  denominator. 

On  inductive  circuit,  however,  x0  appears  in  the  denom- 
inator as  a  term  of  first  order,  and  therefore  constant  poten- 
tial regulation  does  not  take  place  as  well. 


ALTERNATING-CURRENT  GENERATOR.  309 

With  a  non-inductive  external  circuit,  if  the  synchronous 
reactance,  XQ,  of  the  alternator  is  very  large  compared  with 
the  external  resistance,  r, 


current  /=  — 

x 


-g.  1  _E, 


approximately,  or  constant ;  or,  if  the  external  circuit  con- 
tains the  reactance,  x, 

T=-** 1        -  * 


approximately,  or  constant. 

The  terminal  voltage  of  a  non-inductive  circuit  is 


approximately,  or  proportional  to  the  external  resistance. 
In  an  inductive  circuit, 

£° 
x 

approximately,  or  proportional  to  the  external  impedance. 

188.  That  is,  on  a  non-inductive  external  circuit,  an 
alternator  with  very  low  synchronous  reactance  regulates 
for  constant  terminal  voltage,  as  a  constant-potential  ma- 
chine ;  an  alternator  with  a  very  high  synchronous  reac- 
tance regulates  for  a  terminal  voltage  proportional  to  the 
external  resistance,  as  a  constant-current  machine. 

Thus,  every  alternator  acts  as  a  constant-potential  ma- 
chine near  open  circuit,  and  as  a  constant-current  machine 
near  short  circuit.  Between  these  conditions,  there  is  a 
range  where  the  alternator  regulates  approximately  as  a 
constant  power  machine,  that  is  current  and  E.M.F.  vary 
in  inverse  proportion,  as  between  130  and  200  amperes  in 
Fig.  129. 

The  modern  alternators  are  generally  more  or  less  ma- 


310  ALTERNATING-CURRENT  PHENOMENA. 

chines  of  the  first  class  ;  the  old  alternators,  as  built  by 
Jablockkoff,  Gramme,  etc.,  were  machines  of  the  second 
class,  used  for  arc  lighting,  where  constant-current  regula- 
tion is  an  advantage. 

Obviously,  large  external  reactances  cause  the  same  reg- 
ulation for  constant  current  independently  of  the  resistance, 
r,  as  a  large  internal  reactance,  .r0. 

On  non-inductive  circuit,  if 


theoutputis 


hence,  if 
or 


then 

dr 

That  is,  the  power  is  a  maximum,  and 

£ 

and 


7  = 


V2  So  {so  +  r0) 

Therefore,  with  an  external  resistance  equal  to  the  inter- 
nal impedance,  or,  r  —  ^0  =  VV02  +  x^ ,  the  output  of  an 
alternator  is  a  maximum,  and  near  this  point  it  regulates 
for  constant  output ;  that  is,  an  mcrease  of  current  causes 
a  proportional  decrease  of  terminal  voltage,  and  inversely. 

The  field  characteristic  of  the  alternator  shows  this 
effect  plainly. 


SYNCHRONIZING  ALTERNATORS.  311 


CHAPTER    XVIII. 

SYNCHRONIZING    ALTERNATORS. 

189.  All  alternators,  when  brought  to  synchronism  with 
each  other,  will  operate  in  parallel  more  or  less  satisfactorily. 
This  is  due  to  the  reversibility  of  the  alternating-current 
machine  ;  that  is,  its  ability  to  operate  as  synchronous  motor. 
In  consequence  thereof,  if  the  driving  power  of  one  of  sev- 
eral parallel-operating  generators  is  withdrawn,  this  gene- 
rator will  keep  revolving  in  synchronism  as  a  synchronous 
motor ;  and  the  power  with  which  it  tends  to  remain   in 
synchronism  is  the  maximum  power  which  it  can  furnish 
as  synchronous  motor  under  the  conditions  of  running. 

190.  The  principal  and  foremost  condition  of  parallel 
operation  of  alternators  is  equality  of  frequency ;  that  is, 
the  transmission  of  power  from  the  prime  movers  to  the 
alternators  must  be  such  as  to  allow  them  to  run  at  the 
same  frequency  without   slippage  or  excessive  strains   on 
the  belts  or  transmission  devices. 

Rigid  mechanical  connection  of  the  alternators  cannot  be 
considered  as  synchronizing ;  since  it  allows  no  flexibility  or 
phase  adjustment  between  the  alternators,  but  makes  them 
essentially  one  machine.  If  connected  in  parallel,  a  differ- 
ence in  the  field  excitation,  and  thus  the  induced  E.M.F.  of 
the  machines,  must  cause  large  cross-current ;  since  it  cannot 
be  taken  care  of  by  phase  adjustment  of  the  machines. 

Thus  rigid  mechanical  connection  is  not  desirable  for 
parallel  operation  of  alternators. 

191.  The  second  important  condition  of  parallel  opera- 
tion is  uniformity  of  speed  ;  that  is,  constancy  of  frequency. 


312  ALTERNATING-CURRENT  PHENOMENA. 

If,  for  instance,  two  alternators  are  driven  by  independent 
single-cylinder  engines,  and  the  cranks  of  the  engines  hap- 
pen to  be  crossed,  the  one  engine  will  pull,  while  the  other 
is  near  the  dead-point,  and  conversely.  Consequently,  alter- 
nately the  one  alternator  will  tend  to  speed  up  and  the 
other  slow  down,  then  the  other  speed  up  and  the  first 
slow  down.  This  effect,  if  not  taken  care  of  by  fly-wheel 
capacity,  causes  a  "hunting"  or  pumping  action;  that  is,  a 
fluctuation  of  the  lights  with  the  period  of  the  engine  revo- 
lution, due  to  the  alternating  transfer  of  the  load  from  one 
engine  to  the  other,  which  may  even  become  so  excessive 
as  to  throw  the  machines  out  of  step,  especially  when  by  an 
approximate  coincidence  of  the  period  of  engine  impulses 
(or  a  multiple  thereof),  with  the  natural  period  of  oscillation 
of  the  revolving  structure,  the  effect  is  made  cumulative. 
This  difficulty  as  a  rule  does  not  exist  with  turbine  or  water- 
wheel  driving. 

192.  In  synchronizing  alternators,  we  have  to  distin- 
guish the  phenomena  taking  place  when  throwing  the  ma- 
chines in  parallel  or  out  of  parallel,  and  the  phenomena 
when  running  in  synchronism. 

When  connecting  alternators  in  parallel,  they  are  first 
brought  approximately  to  the  same  frequency  and  same 
voltage ;  and  then,  at  the  moment  of  approximate  equality 
of  phase,  as  shown  by  a  phase-lamp  or  other  device,  they 
are  thrown  in  parallel. 

Equality  of  voltage  is  much  less  important  with  modern 
alternators  than  equality  of  frequency,  and  equality  of  phase 
is  usually  of  importance  only  in  avoiding  an  instantaneous 
flickering  of  the  lights  on  the  system.  When  two  alter- 
nators are  thrown  together,  currents  pass  between  the 
machines,  which  accelerate  the  one  and  retard  the  other 
machine  until  equal  frequency  and  proper  phase  relation 
are  reached. 

With  modern  ironclad  alternators,  this  interchange  of 
mechanical  power  is  usually,  even  without  very  careful 


SYNCHRONIZING  ALTERNATORS.  313 

adjustment  before  synchronizing,  sufficiently  limited  net 
to  endanger  the  machines  mechanically  ;  since  the  cross- 
currents, and  thus  the  interchange  of  power,  are  limited 
by  self-induction  and  armature  reaction1. 

In  machines  of  very  low  armature  reaction,  that  is, 
machines  of  "  very  good  constant  potential  regulation," 
much  greater  care  has  to  be  exerted  in  the  adjustment 
to  equality  of  frequency,  voltage,  and  phase,  or  the  inter- 
change of  current  may  become  so  large  as  to  destroy  the 
machine  by  the  mechanical  shock ;  and  sometimes  the 
machines  are  so  sensitive  in  this  respect  that  it  is  prefer- 
able not  to  operate  them  in  parallel.  The  same  applies 
in  getting  out  of  step. 

193.  When   running  in  synchronism,  nearly  all   types 
of  machines  will  operate  satisfactorily ;    a  medium  amount 
of  armature  reaction  is  preferable,  however,  such  as  is  given 
by    modern    alternators  —  not    too    high    to    reduce    the 
synchronizing  power  too  much,  nor  too   low  to  make  the 
machine  unsafe  in  case  of  accident,  such  as  falling  out  of 
step,  etc. 

If  the  armature  reaction  is  very  low,  an  accident,  —  such 
as  a  short  circuit,  falling  out  of  step,  opening  of  the  field 
circuit,  etc.,  —  may  destroy  the  machine.  If  the  armature 
reaction  is  very  high,  the  driving-power  has  to  be  adjusted 
very  carefully  to  constancy  ;  since  the  synchronizing  power 
of  the  alternators  is  too  weak  to  hold  them  in  step,  and 
carry  them  over  irregularities  of  the  driving-power. 

194.  Series  operation  of  alternators  is  possible  only  by 
rigid  mechanical   connection,   or  by  some  means  whereby 
the    machines,   with  regard  to  their  synchronizing  power, 
act  essentially  in  parallel ;  as,  for  instance,  by  the  arrange- 
ment shown  in  Fig.  120,  where  the  two  alternators,  Al}  A2, 
are  connected  in  series,  but   interlinked  by  the  two  coils 
of  a  large  transformer,   T,  of  which  the  one  is  connected 


314 


AL  TERNA  TING-CURRENT  PHENOMENA. 


across  the  terminals  of  one  alternator,  and  the  other  across 
the  terminals  of  the  other  alternator  in  such  a  way  that, 
when  operating  in  series,  the  coils  of  the  transformer  will 


Fig.  136. 

be  without  current.  In  this  case,  by  interchange  of  power 
through  the  transformers,  the  series  connection  will  be 
maintained  stable. 

195.    In  two  parallel  operating  alternators,  as  shown  in 
Fig.  137,  let  the  voltage  at  the  common  bus  bars  be  assumed 


Fig.   137. 


as  zero  line,   or  real  axis  of  coordinates  of   the  complex 
representation  ;  and  let  — 


SYNCHRONIZING  ALTERNATORS.  315 

e    =  difference  of  potential  at  the  common  bus  bars  of 

the  two  alternators, 

Z  =  r  —  jx  =  impedance  of  external  circuit, 
Y  =  g  -\-jb  =  admittance  of  external  circuit  ; 

hence,  the  current  in  external  circuit  is 


Let 

J?i  =  e-i  —  je\  =  #2  (cos  u>1  —  j  sin  £>i)  =  induced  E.M.F.  of  first 

machine  ; 
£2  =  e.2  —  _/>•/  =  a2  (cos  w2  —  j  sin  w2)  =  induced  E.M.F.  of  sec- 

ond machine  ; 

/!    =  /!  -f-//i'  =  current  of  first  machine  ; 
/2    =  /2  -j-yY2'  =  current  of  second  machine  ; 
Z^  =  T!  —  jxi  =  internal  impedance,  and  Yv  =  gi  -\-  jbl  =  inter- 

nal admittance,  of  first  machine  ; 
Z2  =  r2  —  jxz  =  internal  impedance,  and  K2  =gz  ~\~  jb<i  =  inter- 

nal admittance,  of  second  machine. 

Then, 


i^!  ,  or  ^  —je^=  (e 
2Z2,  or  <?2  —jej=  (e 
72  ,  or 


This  gives  the  equations  — 


4*  +  *"-**; 

or  eight  equations  with  nine  variables:  ^,  ^',  ^2,  ^/,  /lf 


316  ALTERNATING-CURRENT  PHENOMENA. 

Combining  these  equations  by  twos, 

elrl  -f  eSxj.  =  er^  +  t\2l2- 
e*r9  +  ^/^2  =  e 
substituted  in 

'i  +  H  = 
we  have 


and  analogously, 

'1^1  —  ^iVi  +  'a  *a  —  <?aVa  =  '  (^  +  ^2  + 
dividing, 


b  +  ^i  +  ^2      ^i  ^;i  +  <?a  ^  —  ^iVi  —  ^a'  ^2  ' 
substituting 

g  =  V  COS  a  Cl    =  tfj  COS  Wj  ^2   =  ^2  COS  d)2 

^  =  z/  sin  a         ^/  =  ^  sin  oJj         ^2'  =  a2  sin  <o2 
gives 

a\  v\  cos  (en  —  aQ  +  a2z>2  cos  (a2  —  a2) 
tfj  z/!  sin  (ai  —  w^)  -\-  a^Vs  sin  (a2  —  a>2) 

as  the  equation  between  the  phase  displacement  angles 
and  oi2  in  parallel  operation. 

The  power  supplied  to  the  external  circuit  is 


of  which  that  supplied  by  the  first  machine  is, 

/i  =  «\  ; 
by  the  second  machine, 

/2  =  «a  • 

The  total  electrical  work  done  by  both  machines  is, 

P  =  Pl  +  P*, 
of.  which  that  done  by  the  first  machine  is, 

PI  =  '!  h  -  e,'  //  ; 
by  the  second  machine, 


SYNCHRONIZING  ALTERNATORS.          .  317 

The  difference  of  output  of  the  two  machines  is, 

denoting 

£>!  -f-  0)2 <QI  —  o>2  s 

~2~  ~2~ 

A^>/AS  may  be  called  the  synchronizing  power  of  the 
machines,  or  the  power  which  is  transferred  from  one  ma- 
chine to.  the  other  by  a  change  of  the  relative  phase  angle. 

196.  SPECIAL  CASE.  —  Two  equal  alternators  of  equaL 
excitation. 


Substituting  this  in  the  eight  initial  equations,  these 
assume  the  form,  — 


e-    =  t  x0  —  t  r0 
e2'  =  /2  .r0  —  //  r0  . 
*g=i\  +'a 
eb  =  i{  +  /a' 

4*  +  4"  -</  +  *"-< 

Combining  these  equations  by  twos, 


substituting  el  =  a  cos  o^ 

e{  =  a  sin  o^ 
^2  =  a  cos  0)2 
e2f  =  a  sin  o)2, 

we  have       a  (cos  wx  +  cos  wa)  =  *  (2  +  r0^  + 
a  (sin  Si  +  sin  w2)  =  e  (x^g  —  r0  li) 

expanding  and  substituting  — 


8  = 


318  AL  TERN  A  TING-CURRENT  PHENOMENA. 


a  cos  e  cos  8  =  e  (  1  + 


rQg -\-Xzb 


a  sin  e  cos  8  =  ^^ ^ 


hence 

That  is 

-and         cos  8  =  - 


tan  e  =  — ^ ^ —  =  constant. 


+  A2  =  constant; 


-M1*5 


z±aiy ,  /^o^-^o^\2. 

cos  8 


at  no-phase  displacement  between  the  alternators,  or, 
-we  have    e  =  ^  — . 


V/(' 


+ 


n>^  +  -*o^\2    ,    fxn  £—  r*b 


From  the  eight  initial  equations  we  get,  by  combina- 


(''o2 


subtracted  and  expanded  — 


.or,  since 

<?!  —  <?2  =  ^  (cos  wj  —  cos  G2)  =  —  2  tf2  sin  c  sin  8 
^/  —  <?/  =  a  (sin  wx  —  sin  w2)  =       2  a  cos  e  sin  8  ; 

we  have 

2  a  s*n  8  -  r0sin  c} 


—  2  ay0  sin  8  cos  (c  -f  a), 


where 


tan  d  =  ±2- . 

/"o 


SYNCHRONIZING  ALTERNATORS.  319 

The  difference  of  output  of  the  two  alternators  is 

A/  =/!  —  /2  =  e  (/i  —  /2)  ; 
hence,  substituting, 


substituting, 


2ggsin8{jfrcos  £  -  r0  sin  c}; 


, 
H 


XQ£  —  r*b 

2 


i    'of,  T  •*<) "  \    i 

2         J  +  V         2 
we  have, 

2a2  sin  8  cos  8  j  *0(  1  +  r°*  +  x°*\  —  r0 


expanding, 

A/  =  nr 


Hence,  the  transfer  of  power  between  the  alternators, 
A  pt  is  a  maximum,  if  8  =  45°  ;  or  Wj  —  w2  =  90°  ;  that  is, 
when  the  alternators  are  in  quadrature. 


320  ALTERNATING-CURRENT  PHENOMENA. 

The   synchronizing  power,   A  p  /  A  8,  is  a  maximum  if 
8  =  0  ;  that  is,  the  alternators  are  in  phase  with  each  other. 

197.   As  an  instance,  curves  may  be  plotted 
for, 

a    =2500, 


with  the  angle  8  =  U)l       a>2  as  abscissae,  giving 

the  value  of  terminal  voltage,  e  • 

the  value  of  current  in  the  external  circuit,  /  =  ey  ; 

the  value  of  interchange  of   current  between  the  alternators, 

*i-*2; 
the  value  of  interchange  of  power  between  the  alternators,  A  p 

=A-/2; 

the  value  of  synchronizing  power,  —  ^  . 

A  o 

For  the  condition  of  external  circuit, 

g  =  0,  b  =  0,  y  =  0, 

.05,  0,  .05, 

.08,  0,  .08, 

.03,  +  .04,  .05, 

.03,  -  .04,  .05. 


SYNCHRONOUS  MOTOR.  321 


CHAPTER   XIX. 

SYNCHRONOUS    MOTOR. 

198.  In  the  chapter  on  synchronizing  alternators  we 
have  seen  that  when  an  alternator  running  in  synchronism 
is  connected  with  a  system  of  given  E.M.F.,  the  work  done 
by  the  alternator  can  be  either  positive  or  negative.  In 
the  latter  case  the  alternator  consumes  electrical,  and 
consequently  produces  mechanical,  power ;  that  is,  runs 
as  a  synchronous  motor,  so  that  the  investigation  of  the 
synchronous  motor  is  already  contained  essentially  in  the 
equations  of  parallel-running  alternators. 

Since  in  the  foregoing  we  have  made  use  mostly  of 
the  symbolic  method,  we  may  in  the  following,  as  an 
instance  of  the  graphical  method,  treat  the  action  of  the 
synchronous  motor  diagrammatically. 

Let  an  alternator  of  the  E.M.F.,  E±,  be  connected  as 
synchronous  motor  with  a  supply  circuit  of  E.M.F.,  EQ, 
by  a  circuit  of  the  impedance  Z. 

If  E0  is  the  E.M.F.  impressed  upon  the  motor  termi- 
nals, Z  is  the  impedance  of  the  motor  of  induced  E.M.F., 
E±.  If  E0  is  the  E.M.F.  at  the  generator  terminals,  Z  is 
the  impedance  of  motor  and  line,  including  transformers 
and  other  intermediate  apparatus.  If  EQ  is  the  induced 
E.M.F.  of  the  generator,  Z  is  the  sum  of  the  impedances 
of  motor,  line,  and  generator,  and  thus  we  have  the  prob- 
lem, generator  of  induced  E.M.F.  EQ,  and  motor  of  induced' 
E.M.F.  El;  or,  more  general,  two  alternators  of  induced 
E.M.Fs.,  E0,  Elf  connected  together  into  a  circuit  of  total 
impedance,  Z. 

Since  in  this  case  several  E.M.Fs.  are  acting  in  circuit 


322  ALTERNATING-CURRENT  PHENOMENA. 

with  the  same  current,  it  is  convenient  to  use  the  current, 
/,  as  zero  line  OI  of  the  polar  diagram.  Fig.  188. 

If  I=i=  current,  and  Z  =  impedance,  r  =  effective 
resistance,  x  =  effective  reactance,  and  s  =  Vr2  -f  x2  = 
absolute  value  of  impedance,  then  the  E.M.F.  consumed 
by  the  resistance  is  E,,  =  ri,  and  in  phase  with  the  cur- 
rent, hence  represented  by  vector  OE,,  ;  and  the  E.M.F. 
consumed  by  the  reactance  is  E2  =  xi,  and  90°  ahead  of 
the  current,  hence  the  E.M.F.  consumed  by  the  impedance 
is  E  =  V(£,,)2  +  (E2f,  or  =  i  Vr2  +  x*  =  is,  and  ahead  of 
the  current  by  the  angle  8,  where  tan  8  =  x  /  r. 

We  have  now  acting  in  circuit  the  E.M.Fs.,  E,  Elf  EQ; 
or  El  and  E  are  components  of  EQ  ;  that  is,  EQ  is  the 
diagonal  of  a  parallelogram,  with  El  and  E  as  sides. 

Since  the  E.M.Fs.  Elf  Ez,  E,  are  represented  in  the 
diagram,  Fig.  138,  by  the  vectors  OE~lf  OE2,  OE,  to  get 
the  parallelogram  of  £Q,  Elt  E,  we  draw  arcs  of  circles 
around  0  with  EQ  ,  and  around  E  with  El  .  Their  point  of 
intersection  gives  the  impressed  E.M.F.,  OEQ  =  EQ,  and 
completing  the  parallelogram  OE  EQ  E±  we  get,  OE±  =  E±  , 
the  induced  E.M.F.  of  the  motor. 


IOE0  is  the  difference  of  phase  between  current  and  im- 
pressed E.M.F.,  or  induced  E.M.F.  of  the  generator. 

IOEi  is  the  difference  of  phase  between  current  and  in- 

duced E.M.F.  of  the  motor. 

And  the  power  is  the  current  /times  the  projection  of  the  E.M.F. 
upon  the  current,  or  the  zero  line  OI. 

Hence,  dropping  perpendiculars,  E^EJ  and  E^E^,  from 
EQ  and  E!  upon  OI,  it  is  — 

P0  =  iX  OE^  =  power  supplied  by  induced   E.M.F.  of  gen- 

erator. 
PI  =  /  X  OE^  =    electric   power  transformed   in   mechanical 

power  by  the  motor. 
P  =  /  x  OEl  =  power  consumed   in   the   circuit   by   effective 

resistance. 


SYNCHRONOUS  MOTOR. 


323 


Since  the  circles  drawn  with  EQ  and  E±  around  O  and  K 
respectively  intersect  twice,  two  diagrams  exist.  In  gen- 
eral, in  one  of  these  diagrams  shown  in  Fig.  138  in  drawn 


Fig.  138. 

lines,  current  and  E.M.F.  are  in  the  same  direction,  repre- 
senting mechanical  work  done  by  the  machine  as  motor- 
In  the  other,  shown  in  dotted  lines,  current  and  E.M.F.  are 
in  opposite  direction,  representing  mechanical  work  con- 
sumed by  the  machine  as  generator. 

Under  certain  conditions,  however,  £Q  is  in  the  same,  E^ 
in  opposite  direction,  with  the  current ;  that  is,  both  ma- 
chines are  generators. 

199.  It  is  seen  that  in  these  diagrams  the  E.M.Fs.  are- 
considered  from  the  point  of  view  of  the  motor ;  that  is,. 


324 


ALTERNATING-CURRENT  PHENOMENA. 


work  done  as  synchronous  motor  is  considered  as  positive, 
work  done  as  generator  is  negative.  In  the  chapter  on  syn- 
chronizing generators  we  took  the  opposite  view,  from  the 
generator  side. 

In  a  single  unit-power  transmission,  that  is,  one  generator 
supplying  one  synchronous  motor  over  a  line,  the  E.M.F. 
consumed  by  the  impedance,  E  =  OE,  Figs.  139  to  141,  con- 
sists of  three  components ;  the  E.M.F.  OE£  —  Ez,  consumed 


Fig.   139. 

by  the  impedance  of  the  motor,  the  E.M.F. 
consumed  by  the  impedance  of  the  line,  and  the  E.M.F. 
EZ  E  =  E±  consumed  by  the  impedance  of  the  generator. 
Hence,  dividing  the  opposite  side  of  the  parallelogram  E1E(), 
in  the  same  way,  we  have :  OEl  =  E1  =  induced  E.M.F.  of 
the  motor,  OEZ  =  2?a  =  E.M.F.  at  motor  terminals  or  at 
end  of  line,  OE3  =  E3  =  E.M.F.  at  generator  terminals, 
or  at  beginning  of  line.  OEQ  =  EQ  =  induced  E.M.F.  of 
generator. 


SYNCHRONOUS  MOTOR. 


325 


The  phase  relation  of  the  current  with  the  E.M.Fs.  £lt 
,  depends  upon  the  current  strength  and  the  E.M.Fs.  El 


and 


200.  Figs.  139  to  141  show  several  such  diagrams  for 
different  values  of  Elf  but  the  same  value  of  /  and  EQ. 
The  motor  diagram  being  given  in  drawn  line,  the  genera- 
tor diagram  in  dotted  line. 


Fig.  140. 

As  seen,  for  small  values  of  E1  the  potential  drops  in 
the  alternator  and  in  the  line.  For  the  value  of  E1  =  E0 
the  potential  rises  in  the  generator,  drops  in  the  line,  and 
rises  again  in  the  motor.  For  larger  values  of  Ely  thfe 
potential  rises  in  the  alternator  as  well  as  in  the  line,  so 
that  the  highest  potential  is  the  induced  E.M.F.  of  the 
motor,  the  lowest  potential  the  induced  E.M.F.  of  the  gen- 
erator. 


326 


ALTERNATING-CURRENT  PHENOMENA, 


It  is  of  interest  now  to  investigate  how  the  values  of 
these  quantities  change  with  a  change  of  the  constants. 


Fig.  747. 

201.   A.  —  Constant  impressed  E.M.F.  Ev,  constant  current 
strength  I  =  i,  variable  motor  excitation  Ev      (Fig.  142.) 

If  the  current  is  constant,  =  z;  OE,  the  E.M.F.  con- 
sumed by  the  impedance,  and  therefore  point  E,  are  con- 
stant. Since  the  intensity,  but  not  the  phase  of  EQ  is 
constant,  EQ  lies  on  a  circle  eQ  with  EQ  as  radius.  From 
the  parallelogram,  OE  EQ  El  follows,  since  E1 EQ  parallel 
and  =  OE,  that  El  lies  on  a  circle  el  congruent  to  the  circle 
eQ,  but  with  Ei}  the  image  of  E,  as  center  :  OEi  =  OE. 

We  can  construct  now  the  variation  of  the  diagram  with 
the  variation  of  El ;  in  the  parallelogram  OE  EQ  E1 ,  O  and 
E  are  fixed,  and  E0  and  El  move  on  the  circles  <?0  el  so  that 
EQ  E^  is  parallel  to  OE. 


SYNCHRONOUS  MOTOR. 


327 


The  smallest  value  of  El  consistent  with  current  strength 
/  is  Olj  =  E^,  01  =  EQ.  In  this  case  the  power  of  the 
motor  is  Olj1  x  /,  hence  already  considerable.  Increasing 
El  to  02"^  OSj,  etc.,  the  impressed  E.M.Fs.  move  to  02,  03, 
etc.,  the  power  is  /  x  02^,  I  x  03^,  etc.,  increases  first, 


Fig.  142. 

reaches  the  maximum  at  the  point  3j,  3,  the  most  extreme 
point  at  the  right,  with  the  impressed  E.M.F.  in  phase  with 
the  current,  and  then  decreases  again,  while  the  induced 
E.M.F.  of  the  motor  E^  increases  and  becomes  =  £Q  at 
4,,  4.  At  515  5,  the  power  becomes  zero,  and  further  on 
negative  ;  that  is,  the  motor  has  changed  to  a  dynamo,  and 


328  AL  TERNA  TING-CURRENT  PHENOMENA. 

produces  electrical  energy,  while  the  impressed  E.M.F.  E^ 
still  furnishes  electrical  energy,  that  is,  both  machines  as 
generators  feed  into  the  line,  until  at  61}  6,  the  power  of  the 
impressed  E.M.F.  E§  becomes  zero,  and  further  on  power 
begins  to  flow  back ;  that  is,  the  motor  is  changed  to  a  gen- 
erator and  the  generator  to  a  motor,  and  we  are  on  the 
generator  side  of  the  diagram.  At  1l,  7,  the  maximum  value 
of  Elt  consistent  with  the  current  /,  has  been  reached,  and 
passing  still  further  the  E.M.F.  El  decreases  again,  while 
the  power  still  increases  up  to  the  maximum  at  Slt  8,  and 
then  decreases  again,  but  still  El  remaining  generator,  EQ 
motor,  until  at  11^  11,  the  power  of  EQ  becomes  zero;  that 
is,  EQ  changes  again  to  a  generator,  and  both  machines  are 
generators,  up  to  12lf  12,  where  the  power  of  El  is  zero,  El 
changes  from  generator  to  motor,  and  we  come  again  to 
the  motor  side  of  the  diagram,  and  while  El  still  decreases, 
the  power  of  the  motor  increases  until  lu  1,  is  reached. 

Hence,  there  are  two  regions,  for  very  large  El  from 
5  to  6,  and  for  very  small  El  from  11  to  12,  where  both 
machines  are  generators ;  otherwise  the  one  is  generator, 
the  other  motor. 

For  small  values  of  El  the  current  is  lagging,  begins, 
however,  at  2  to  lead  the  induced  E.M.F.  of  the  motor  Elf 
at  3  the  induced  E.M.F.  of  the  generator  E0. 

It  is  of  interest  to  note  that  at  the  smallest  possible 
value  of  EI}  lj,  the  power  is  already  considerable.  Hence, 
the  motor  can  run  under  these  conditions  only  at  a  certain 
load.  If  this  load  is  thrown  off,  the  motor  cannot  run  with 
the  same  current,  but  the  current  must  increase.  We  have 
here  the  curious  condition  that  loading  the  motor  reduces, 
unloading  increases,  the  current  within  the  range  between 
1  and  12. 

The  condition  of  maximum  output  is  3,  current  in  phase 
with  impressed  E.M.F.  Since  at  constant  current  the  loss 
is  constant,  this  is  at  the  same  time  the  condition  of  max- 
imum efficiency :  no  displacement  of  phase  of  the  impressed 


SYNCHRONOUS  MOTOR. 


329 


E.M.F.,  or  self-induction  of  the  circuit  compensated  by  the 
effect  of  the  lead  of  the  motor  current.  This  condition  of 
maximum  efficiency  of  a  circuit  we  have  found  already  in 
the  Chapter  on  Inductance  and  Capacity. 


202.       B.     EQ  and  El  constant,  I  variable. 

Obviously  EQ  lies  again  on  the  circle  eQ  with  EQ  as  radius 
and  O  as  center. 


Fig.  143. 


E  lies  on  a  straight  line  e,  passing  throtigh  the  origin; 

Since  in  the  parallelogram  OE  E0  Ev  EEQ  =  E^  we 
derive  EQ  by  laying  a  line  EEQ  =  E±  from  any  point  E 
in  the  circle  eQ,  and  complete  the  parallelogram. 

All  these  lines  EEQ  envelop  a  certain  curve  elt  which 


030  ALTERNATING-CURRENT  PHENOMENA. 

can  be  considered  as  the  characteristic  curve  of  this  prob- 
lem, just  as  circle  e^  in  the  former  problem. 

These  curves  are  drawn  in  Figs.  143,  144,  145,  for  the 
three  cases  :  1st,  El  =  EQ ;  2d,  El  <  EQ ;  3d,  £1>£Q. 

In  the  first   case,   El  =  EQ  (Fig.  127),  we  see  that  at 


Fig.   144. 


very  small  current,  that  is  very  small  OE,  the  current  / 
leads  the  impressed  E.M.F.  EQ  by  an  angle  EQOf  =  WQ. 
This  lead  decreases  with  increasing  current,  becomes  zero, 
and  afterwards  for  larger  current,  the  current  lags.  Taking 
now  any  pair  of  corresponding  points  E,  EQ,  and  producing 
EEQ  until  it  intersects  eit  in  Eif  we  have  ^^  Ei  OE  —  90°, 
El  =  EQ ,  thus  :  OE1  =  EEQ=OEQ  =  EQEt ;  that  is,  EE{  = 


SYNCHRONOUS  MOTOR. 


331 


2EQ.  That  means  the  characteristic  curve  el  is  the  enve- 
lope of  lines  EEiy  of  constant  lengths  2EQ,  sliding  between 
the  legs  of  the  right  angle  Et  OE;  hence,  it  is  the  sextic 
hypocyloid  osculating  circle  <?0,  which  has  the  general  equa- 
tion, with  e,  ei  as  axes  of  coordinates  : 


In  the  next  case,  E1  <  EQ  (Fig.  144)  we  see  first,  that 
the  current  can   never  become  zero  like  in  the  first  case, 


V 


Fig.  145. 


EI  =  EQ,  but  has  a  minimum  value  corresponding  to  the 
minimum  value  of  OEl :   I{  =  — — ,  and  a  maximum 

value  :  //'  =  — — .    Furthermore,  the  current  can  never 

lead  the  impressed  E.M.F.  E^,  but  always  lags.     The  mini- 


332  ALTERNATING-CURRENT  PHENOMENA. 

mum  lag  is  at  the  point  H.  The  locus  ev  as  envelope  of  the 
lines  EEty  is  a  finite  sextic  curve,  shown  in  Fig.  144. 

If  El  <  EQ ,  at  small  EQ  —  El ,  H  can  be  above  the  zero 
line,  and  a  range  of  leading  current  exist  between  two  ranges 
of  lagging  current. 

In  the  case  E1  >  EQ  (Fig.  145)  the  current  cannot  equal 
zero  either,  but  begins  at  a  finite  value  C±,  corresponding 

to  the  minimum  value  of  OEQ  :  //  =      *     — -.     At  this 

value  however,  the  alternator  E1  is  still  generator  and 
changes  to  a  motor,  its  power  passing  through  zero,  at  the 
point  corresponding  to  the  vertical  tangent,  onto  elf  with 
a  very  large  lead  of  the  impressed  E.M.F.  against  the  cur- 
rent. At  H  the  lead  changes  to  lag. 

The  minimum  and  maximum  value  of  current  in  the 
three  conditions  are  given  by : 

Minimum:  Maximum: 

1st.   7=0,  7=^. 


Since  tfie  current  passing  over  the  line  at  El  =  O,  that 
is,  when  the  motor  stands  still,  is  70  =  EQj  z,  we  see  that 
in  such  a  synchronous  motor-plant,  when  running  at  syn- 
chronism, the  current  can  rise  far  beyond  the  value  it  has 
at  standstill  of  the  motor,  to  twice  this  value  at  1,  some- 
what less  at  2,  but  more  at  3. 

203.    C.    EQ  =  constant,  El  varied  so  that  the  efficiency  is  a 
maximum  for  all  currents.  •    (Fig.  146.) 

Since  we  have  seen  that  the  output  at  a  given  current 
strength,  that  is,  a  given  loss,  is  a  maximum,  and  therefore 


SYNCHRONOUS  MOTOR. 


333 


the  efficiency  a  maximum,  when  the  current  is  in  phase 
with  the  induced  E.M.F.  EQ  of  the  generator,  we  have  as 
the  locus  of  EQ  the  point  EQ  (Fig.  146),  and  when  E  with 
increasing  current  varies  on  <?,  E±  must  vary  on  the  straight 
line  ev  parallel  to  c. 

Hence,  at  no-load  or  zero  current,  El  =  E0,  decreases 
with  increasing  load,  reaches  a  minimum  at  OE^  perpen- 
dicular to  clt  and  then  increases  again,  reaches  once  more 


Fig.   146. 


El  =  EQ  at  E?,  and  then  increases  beyond  E0.  The  cur- 
rent is  always  ahead  of  the  induced  E.M.F.  El  of  the  motor, 
and  by  its  lead  compensates  for  the  self-induction  of  the 
system,  making  the  total  circuit  non-inductive. 

The  power  is  a  maximum  at  Ef,  where  OEf  =  EfEQ  = 
1/2  x  ~OE^  and  is  then  =  /  x  "^7/2.     Hence,  since  OEf  = 


EJ2,f=E()/2randP 


hence  =  the  maxi- 


mum power  which,  over  a  non-inductive  line  of  resistance  r 
can  be  transmitted,  at  50  per  cent,  efficiency,  into  a  non- 
inductive  circuit. 


-334  ALTERNATING-CURRENT  PHENOMENA. 

In  this  case, 


In  general,  it  is,  taken  from  the  diagram,  at  the  condi- 
tion of  maximum  efficiency  : 


Comparing  these  results  with  those  in  Chapter  IX.  on 
Self-induction  and  Capacity,  we  see  that  the  condition  of 
maximum  efficiency  of  the  synchronous  motor  system  is 
the  same  as  in  a  system  containing  only  inductance  and 
•capacity,  the  lead  of  the  current  against  the  induced  E.M.F. 
El  here  acting  in  the  same  way  as  the  condenser  capacity 
in  Chapter  IX. 


204. 


Fig.    147. 


D.    En  =  constant ;  P  =  constant. 


If  the  power  of  a  synchronous  motor  remains  constant, 
we  have  (Fig.  147)  /  x  OE^  =  constant,  or,  since  OE1  — 


SYNCHRONOUS  MOTOR. 


335 


Ir,    I  =  OE1/  r,    and:     OE1  x  OE?  =  O£l  X  E1EJ  = 
constant. 

Hence  we  get  the  diagram  for  any  value  of  the  current 
/,  at  constant  power  Plt  by  making  OE1  =  I  r,  E1E01  =  Pl  j  I 
erecting  in  EQl  a  perpendicular,  which  gives  two  points  of 
intersection  with  circle  eQ,  EQ,  one  leading,  the  other  lagging. 
Hence,  at  a  given  impressed  E.M.F.  EQ,  the  same  power  P± 


E, 


1250  7 

1100/1580  31/16.7 

1480  32 

1050/1840  2/25 

2120 
2170 


37.5 
40 


45.5 


16.7 


Fig.   U8. 

can  be  transmitted  by  the  same  current  I  with  two  different 
induced  E.M.Fs.  E}  of  the  motor;  one,  OEl  =  EEQ  small, 
corresponding  to  a  lagging  current ;  and  the  other,  OEl  = 
EEQ  large,  corresponding  to  a  leading  current.  The  former 
is  shown  in  dotted  lines,  the  latter  in  drawn  lines,  in  the 
diagram,  Fig.  147. 

Hence  a  synchronous  motor  can  work  with  a  given  out- 
put, at  the  same  current  with  two  different  counter  E.M.Fs. 


336 


ALTERNATING-CURRENT  PHENOMENA. 


E1.     In  one  of  the  cases  the  current   is   leading,  in  the 
Dther  lagging. 

In  Figs.  148  to  151  are  shown  diagrams,  giving  the  points 

E0  =  impressed  E.M.F.,  assumed  as  constant  =  1000  volts, 
E  =  E.M.F.  consumed  by  impedance, 
E'  =  E.M.F.  consumed  by  resistance. 


EflOOO 

P=6000 

34O<  E,<1920 

7<  I  <  43 


Fig.  149. 


I 

1450  17.3 

1170/1910     10/30 
1040/1930     8/37.5 


10/30 
17.3 


of  the  motor,  Elt  is  OElt  equal  and 
shown    in   the  diagrams,   to   avoid 


The  counter  E.M.F. 
parallel  EEQ,  but  not 
complication. 

The  four  diagrams  correspond  to  the  values  of  power, 
or  motor  output, 
P  =  1,000,       6,000, 


9,000, 


12,000  watts,  and  give  : 
1  <  I  <  49          Fig.  132. 


P  =    1,000  46  <  El  <  2,200, 

P  =    6,000  340  <  £,  <  1,920,          7  <  I  <  43  Fig.  133. 

P  =    9,000  540  <  El  <  1,750,  11.8  <  /  <  38.2  Fig.  134. 

P  =  12,000  920  <  El  <  1,320,  20     <  I  <  30  Fig.  153. 


SYNCHRONOUS  MOTOR. 


337 


E,      I 

*     1440  21.2 

3     1200/1660  15/30 

1080/1750  13/34.7 


900/1590   11.8/38.2. 


720/1100   13/34.7 
620/820   15/30 
/3     540      21.2 


3        1280  24.5 

2       1120/1320       21/28.6 
all— l-QQO/1260       30/30 


920/1100 
020 


21/28.6 
24.5 


P=I200O 

920<  E,<  1320 

20<l<30 


Fig.  151. 

As  seen,  the  permissible  value  of  counter  E.M.F.  Ev  and 
of  current  /,  becomes  narrower  with  increasing  output. 


338  ALTERNATING-CURRENT  PHENOMENA. 

In  the  diagrams,  different  points  of  EQ  are  marked  with 
1,  2,  3  .  .  . ,  when  corresponding  to  leading  current,  with 
21,  31,  .  .  . ,  when  corresponding  to  lagging  current. 

The  values  of  counter  E.M.F.  Ev  and  of  current  7  are 
noted  on  the  diagrams,  opposite  to  the  corresponding  points 

*o- 

In  this  condition  it  is  interesting  to  plot  the  current  as 

function  of  the  induced  E.M.F.  El  of  the  motor,  for  con- 
stant power  /V  Such  curves  are  given  in  Fig.  155  and 
explained  in  the  following  on  page  345. 

205.  While  the  graphic  method  is  very  convenient  to 
get  a  clear  insight  into  the  interdependence  of  the  different 
quantities,  for  numerical  calculation  it  is  preferable  to  ex- 
press the  diagrams  analytically. 

For  this  purpose, 

Let  z  =  Vr2  -j-  x2  =  impedance  of  the  circuit  of  (equivalent) 
resistance  r  and  (equivalent)  reactance  x  =  2  TT  NL,  containing 
the  impressed  E.M.F.  e0*  and  the  counter  E.M.F.  et  of  the  syn- 
chronous motor;  that  is,  the  E.M.F.  induced  in  the  motor  arma- 
ture by  its  rotation  through  the  (resultant)  magnetic  field. 

Let  i  =  current  in  the  circuit  (effective  values). 

The  mechanical  power  delivered  by  the  synchronous 
motor  (including  friction  and  core  loss)  is  the  electric 
power  consumed  by  the  C. E.M.F.  e1;  hence  — 

p  =  *>!  cos  ft,^),  (1) 

thus,  — 


*  If  f0  =  E.M.F.  at  motor  terminals,  z  =  internal  impedance  of  the 
motor;  if  eo=  terminal  voltage  of  the  generator,  z  =  total  impedance  of  line 
and  motor;  if  t0=  E.M.F.  of  generator,  that  is,  E.M.F.  induced  in  generator 
armature  by  its  rotation  through  the  magnetic  field,  z  includes  the  generator 
impedance  also. 


SYNCHRONOUS  MOTOR.  339 

The  displacement  of  phase  between  current  i  and  E.M.F. 
=  z  i  consumed  by  the  impedance  z  is  : 


cos  (ie)  =  - 


sin  (/<?) 


x 


(3) 


Since  the  three  E.M.Fs.  acting  in  the  closed  circuit : 

e0  =  E.M.F.  of  generator, 

fi  =  C.E.M.F.  of  synchronous  motor, 

e  =  zi  =  E.M.F.  consumed  by  impedance, 

form  a  triangle,  that  is,  c^  and  e  are  components  of  ^0,  it  is 
(Fig.  152) : 

e1   „  2   eZ  .1  „?.   ^2  ,'2 

hence,      cos  (*,.#)  =  •*- —       —  =  -0 — - — .  (5) 

2  e^e  '2,zie^ 

since,  however,  by  diagram  : 

cos  (el ,  e)  =  cos  (/,  e  —  /',  e^) 

=  cos  (/,  e)  cos  (/,  ^i)  +  sin  (t,  e)  sin  (/,  ^)        (6) 

substitution  of  (2),  (3)  and  (5)  in  (6)  gives,  after  some  trans- 
position : 

the  Fundamental  Equation  of  tJie  Synchronous  Motor,  relat- 
ing impressed  E.M.F.,  <?0 ;  C. E.M.F.,  ^  ;  current  z;  power, 
/,  and  resistance,  r ;  reactance,  x ;  impedance  s. 

This  equation  shows  that,  at  given  impressed  E.M.F.  e$f 
and  given  impedance  s  =  Vr2  +  x*,  three  variables  are  left, 
ev  i,p,  of  which  two  are  independent.  Hence,  at  given  ^ 
and  s,  the  current  i  is  not  determined  by  the  load  /  only, 
but  also  by  the  excitation,  and  thus  the  same  current  i  can 
represent  widely  different  loads  p,  according  to  the  excita- 
tion ;  and  with  the  same  load,  the  current  i  can  be  varied 
in  a  wide  range,  by  varying  the  field  excitation  e1. 

The  meaning  of  equation  (7)  is  made  more  perspicuous 


340  ALTERNATING-CURRENT  PHENOMENA. 


by  some  transformations,  which  separate  ev  and  i,  as  func- 
tion of/  and  of  an  angular  parameter  <£. 
Substituting  in  (7)  the  new  coordinates  : 


V2 


V2 


or, 


_ 

V2 


we  get 


substituting  again,         e<f  =  a 
Izp  =  b 

r  =  €Z 
hence,  x  =  z  Vl  —  e2 


jr.  753. 


we  jret 


a  —  a  V2  —  e  b  =  V(l  —  e2)  (2  a2  —  2  £2  - 
and,  squared, 

substituting 

gives,  after  some  transposition, 

v*  -f  ze/2  =  (-1  ~  *")  a  (a  —  2  tb\ 


(9) 


)»         (11) 

—  0,  (12) 

(13) 
(14) 


SYNCHRONOUS  MOTOR.  341 

hence'if 


i*  +  w*  =  £*  (16) 

the  equation  of  a  circle  with  radius  R. 

Substituting  now  backwards,  we  get,  with  some  trans- 
positions : 

{r*  (ef  +  z*i2)  -  z*  (Vo2  -  2  r/)}2  +  {r  x  (e?  -  z*i2)}2  = 

*2.sV(^02-4r/)  (17) 

the  Fundamental  Rquation  of  the  Synchronous  Motor  in  a 
modified  form. 

The  separation  of  e±  and  i  can  be  effected  by  the  intro- 
duction of  a  parameter  <£  by  the  equations  : 

r3-  (e?  —  z2  /2)  -  z2  (ef  —  2rp)=xze()  V<r0a  —  ±rp  cos  <£ 


rx  (e?  -  z2/2)  =xze»  Vtf  -  4  r/  sin    '   l     ' 
These  equations  (18),  transposed,  give 

N 

+  sin</> 


The  parameter  <^>  has  no  direct  physical  meaning,  appar- 
ently. 

These  equations  (19)  and  (20),  by  giving  the  values  ef 
el  and  i  as  functions  of  /  and  the  parameter  <£  enable  us 
to  construct  the  Power  Characteristics  of  the  Synchronous 
Motor,  as  the  curves  relating  ev  and  i,  for  a  given  power  /, 
by  attributing  to  <£  all  different  values. 


342  ALTERNATING-CURRENT  PHENOMENA. 

Since  the  variables  v  and  w  in  the  equation  of  the  circle 
(16)  are  quadratic  functions  of  e1  and  /',  the  Power  Charac- 
teristics of  the  Synchronous  Motor  are  Quartic  Curves. 

They  represent  the  action  of  the  synchronous  motor 
under  all  conditions  of  load  and  excitation,  as  an  element 
of  power  transmission  even  including  the  line,  etc. 

Before  discussing  further  these  Power  Characteristics, 
some  special  conditions  may  be  considered. 

206.  A.     Maximum   Output. 

Since  the  expression  of  el  and  i  [equations  (19)  and 
(20)]  contain  the  square  root,  W02  —  4  rp,  it  is  obvious 
that  the  maximum  value  of  /  corresponds  to  the  moment 
where  this  square  root  disappears  by  passing  from  real  to 
imaginary  ;  that  is, 

tf  _  4  rp  =  0, 

°r> 

/  =  £..  (21) 

This  is  the  same  value  which  represents  the  maximum 
power  transmissible  by  E.M.F.,  eQ,  over  a  non-inductive  line 
of  resistance,  r\  or,  more  generally,  the  maximum  power 
which  can  be  transmitted  over  a  line  of  impedance, 

into  any  circuit,  shunted  by  a  condenser  of  suitable  capacity. 
Substituting  (21)  in  (19)  and  (20),  we  get, 


and  the  displacement  of  phase  in  the  synchronous  motor. 

cor(A,0-^--i 

tc±       z 

hence, 

tan  fa,  /)  =  -?,  (23) 


SYNCHRONOUS  MOTOR.  343 

that  is,  the  angle  of  internal  displacement  in  the  synchron- 
ous motor  i§  equal,  but  opposite  to,  the  angle  of  displace- 
ment of  line  impedance, 

('i,  0  =  -  (',  0, 

=  ~  <X  '),  (24) 

and  consequently, 

(.-0,0=0;  (25) 

that  is,  the  current,  z,  is  in  phase  with  the  impressed 
E.M.F.,  *0. 

If  2  <  2  r,   el  <  <?0;  that  is,  motor  E.M.F.  <  generator  E.M.F. 

If  z  =  2  r,   el  =  e0 ;  that  is,  motor  E.M.F.  =  generator  E.M.F. 

If  z  >  2  r,   <?!  >  r0;  that  is,  motor  E.M.F.  >  generator  E.M.F. 

In  either  case,  the  current  in  the  synchronous  motor  is 
leading. 

207.  B.     Running  Light,  p  =  0. 

When  running  light,  or  for  /  =  0,  we  get,  by  substitut- 
ing in  (19)  and  (20), 


(26) 


Obviously  this  condition  cannot  well  be  fulfilled,  since  p 
must  at  least  equal  the  power  consumed  by  friction,  etc.  ; 
and  thus  the  true  no-load  curve  merely  approaches  the  curve 
/  =  0,  being,  however,  rounded  off,  where  curve  (26)  gives 
sharp  corners. 

Substituting  /  =  0  into  equation  (7)  gives,  after  squar- 
ing and  transposing, 

e*  +  e<*  4-  3*,-«  -  2  ^V  -  2  22rV  +  2  ra*'V  -  2  *2*V  =  0.  (27) 

This  quartic  equation  can  be  resolved  into  the  product 
of  two  quadratic  equations, 

0.  |  (28) 

0.  j 


344  ALTERNATING-CURRENT  PHENOMENA. 

which  are  the  equations  of  two  ellipses,  the  one  the  image 
of  the  other,  both  inclined  with  their  axes. 

The  minimum  value  of  C.E.M.F.,  eit  is  ^  =  0  at  /  =  ^2.  (29) 
The  minimum  value  of  current,  z,  is  /  =  0  at  et  =  e0  .  (30) 
The  maximum  value  of  E.M.F.,  elt  is  given  by  Equation  (28)', 

/=  e*  +  22z2  -e<?±2  xiel  =  0  ; 
by  the  condition, 


hence, 


The  maximum  value  of  current,  z,  is  given  by  equation 
(28)  by 

—  =  0,  as 
del 

(32) 

If,  as  abscissas,  elt  and  as  ordinates,  zi,  are  chosen,  the 
axis  of  these  ellipses  pass  through  the  points  of  maximum 
power  given  by  equation  (22). 

It  is  obvious  thus,  that  in  the  V-shaped  curves  of  syn- 
chronous motors  running  light,  the  two  sides  of  the  curves 
are  not  straight  lines,  as  usually  assumed,  but  arcs  of  ellipses, 
the  one  of  concave,  the  other  of  convex,  curvature. 

These  two  ellipses  are  shown  in  Fig.  154,  and  divide  the 
whole  space  into  six  parts  —  the  two  parts  A  and  A',  whose 
areas  contain  the  quartic  curves  (19)  (20)  of  synchronous 
motor,  the  two  parts  B  and  B',  whose  areas  contain  the 
quartic  curves  of  generator,  and  the  interior  space  C  and 
exterior  space  D,  whose  points  do  not  represent  any  actual 
condition  of  the  alternator  circuit,  but  make  el ,  i  imaginary. 

A  and  A'  and  the  same  B  and  B' ',  are  identical  condi- 
tions of  the  alternator  circuit,  differing  merely  by  a  simul- 


SYNCHRONOUS  MOTOR. 


345 


\ 


r 


\ 


I 


\ 


4000          3000  \^  2000          1000 


Volts  1000          2000\/3000         4000          5000 


\ 


/A' 


\ 


\ 


Fig.  154. 

taneous  reversal  of  current  and  E.M.F.  ;  that  is,  differing 
by  the  time  of  a  half  period. 

Each  of  the  spaces  A  and  B  contains  one  point  of  equa- 
tion (22),  representing  the  condition  of  maximum  output 
of  generator,  viz.,  synchronous  motor. 

208.    C.    Minimum  Current  at  Given  Power. 

The  condition  of  minimum  current,  t,  at  given  power,  /, 
is  determined  by  the  absence  of  a  phase  displacement  at  the 
impressed  E.M.F.  eQ, 


346  AL  TERNA  TING-CURRENT  PHENOMENA. 

This  gives  from  diagram  Fig.  153, 

e1*  =  e(?  +  i*z*-2ie0r,  (33) 

or,  transposed, 

This  quadratic  curve  passes  through  the  point  of  zero 
current  and  zero  power, 

through  the  point  of  maximum  power  (22), 


and  through  the  point  of  maximum  current  and  zero  power, 


enx 


r 


(35) 


and  divides  each  of  the  quartic  curves  or  power  character- 
istics into  two  sections,  one  with  leading,  the  other  with 
lagging,  current,  which  sections  are  separated  by  the  two 
points  of  equation  34,  the  one  corresponding  to  minimum, 
the  other  to  maximum,  current. 

It  is  interesting  to  note  that  at  the  latter  point  the 
current  can  be  many  times  larger  than  the  current  which 
would  pass  through  the  motor  while  at  rest,  which  latter 
current  is, 

/  =  'J2,  (36) 

while  at  no-load,  the  current  can  reach  the  maximum  value, 
/=^,  (35) 

the  same  value  as  would  exist  in  a  non-inductive  circuit  of 
the  same  resistance. 

The  minimum  value  at  C.E.M.F.  el}  at  which  coincidence 


SYNCHRONOUS  MOTOR.  347 

of  phase  (eQ ,  -i)  =  0,  can  still  be  reached,  is  determined  from 
equation  (34)  by, 


as 

i  —  e  -         —     -  (37} 

The  curve  of  no-displacement,  or  of  minimum  current,  is 
shown  in  Figs.  138  and  139  in  dotted  lines.* 

209.      D.    Maximum  Displacement  of  Phase. 

(e%,  i}  =  maximum. 
At  a  given  power/  the  input  is, 

A  =P  +  i*r  =  e,i  cos  (*0,  *)  ;  (38) 

hence, 

cosfo,  0  =  /+/V.  (39) 

At  a  given  power  /,  this  value,  as  function  of  the  current 
i,  is  a  maximum  when 

d_(p  + 

di\ 
this  gives, 


(40) 
or, 

(41) 

That  is,  the  displacement  of  phase,  lead  or  lag,  is  a 
maximum,  when  the  power  of  the  motor  equals  the  power 

*  It  is  interesting  to  note  that  the  equation  (34)  is  similar  to  the  value, 
<?!  =  \/(^0  —  2  r)2  —  z'2jr2,  which  represents  the  output  transmitted  over  an 
inductive  line  of  impedance,  z  =  vV2  +  jr2  into  a  non-inductive  circuit. 

Equation  (34)  is  identical  with  the  equation  giving  the  maximum  voltage, 
e± ,  at  current,  i,  which  can  be  produced  by  shunting  the  receiving  circuit  with  a 

condenser;   that  is,  the  condition  of  "  complete  resonance  "  of  the  line,  z  = 

x 
Vr'2  +  x'2,  with  current,  ».     Hence,  referring  to  equation  (35),  el  =  t0  ~  is 

the  maximum  resonance  voltage  of  the  line,  reached  when  closed  by  a  con- 
denser of  reactance,  —  x. 


348 


ALTERNATING-CURRENT  PHENOMENA. 


consumed  by  the  resistance ;  that  is,  at  the  electrical  effi- 
ciency of  50  per  cent. 

Substituting  (40)  in  equation  (7)  gives,  after  squaring 


/     N 


TSOO        8000^       #WU        3000        3uOO 
Fig.  155. 

and  transposing,  the  Ouartic  Equation  of   Maximum  Dis- 
placement, 

<>02  -  e*y  +  **z2  (s2  +  8  r2)  +  2  j*e*  (5  r2  -  22)  -  2  / V 

(32  +  3  ^  =  Oi  (42) 

The  curve  of  maximum  displacement  is  shown  in  dash- 
dotted  lines  in  Figs.  154  and  155.     It  passes  through  the 


SYNCHRONOUS  MOTOR.  349 

point  of  zero  current  —  as  singular  or  nodal  point  —  and 
through  the  point  of  maximum  power,  where  the  maximum 
displacement  is  zero,  and  it  intersects  the  curve  of  zero 
displacement. 

210.  E.    Constant  Counter  E.M.F. 

At  constant  C.E.M.F.,  el  =  constant, 

If 

the  current  at  no-load  is  not  a  minimum,  and  is  lagging. 
With  increasing  load,  the  lag  decreases,  reaches  a  mini- 
mum, and  then  increases  again,  until  the  motor  falls  out  of 
step,  without  ever  coming  into  coincidence  of  phase. 


If 


the  current  is  lagging  at  no  load ;  with  increasing  load  the 
lag  decreases,  the  current  comes  into  coincidence  of  phase 
with  eQ ,  then  becomes  leading,  reaches  a  maximum  lead  ; 
then  the  lead  decreases  again,  the  current  comes  again  into 
coincidence  of  phase,  and  becomes  lagging,  until  the  motor 
falls  out  of  step. 

If  eQ  <  <?! ,  the  current  is  leading  at  no  load,  and  the 
lead  first  increases,  reaches  a  maximum,  then  decreases  ; 
and  whether  the  current  ever  comes  into  coincidence  of 
phase,  and  then  becomes  lagging,  or  whether  the  motor 
falls  out  of  step  while  the  current  is  still  leading,  depends, 
whether  the  C.E.M.F.  at  the  point  of  maximum  output  is 
>  <?0  or  <  *0. 

211.  F.    Numerical  Instance. 

Figs.  154  and  155  show  the  characteristics  of  a  100- 
kilowatt  motor,  supplied  from  a  2500-volt  generator  over  a 
distance  of  5  miles,  the  line  consisting  of  two  wires,  No. 
2  B.  &  S.G.,  18  inches  apart. 


350  ALTERNATING-CURRENT  PHENOMENA. 

In  this  case  we  have, 

<?0  =  2500  volts  constant  at  generator  terminals;  ^| 
r  —      10  ohms,  including  line  and  motor  ;  /^gs 

x  =      20  ohms,  including  line  and  motor  ;  j 

hence  z  =     22.36  ohms. 

Substituting  these  values,  we  get, 

25002  -  e*  -  500  i*  -  20  /  =  40  V*V  -/2  (7) 

{^2  +  500  ?2  -  31.25  X  106  +  100  /}2  +  (2  ^2  -  1000  /2}2  = 

7.8125  x  1015  -  5  +  109/.  (17) 

el  =  5590  (19) 

V|  {(1  —  3.2  x  10~6/)  +  (.894  cos  <£+  .447sin  <£)  Vl-6.4xlO-6/}. 
*  =  559  (20) 


—  6.4xlO-6/}. 

Maximum  output, 

p  =  156.25  kilowatts  (21) 

at  *i  =  2,795  volts 

i  =  125  amperes 
Running  light, 

^  +  500  /a  -  6.25  x  104  =p  40  /^  =  0 
^  =  20  /  ±  V6.25  X  104  —  100  i* 

At  the"  minimum  value  of  C.E.M.F.  e1  =  0  is  /  =  112  (29) 
At  the  minimum  value  of  current,  /  =  0  is  el  =  2500  (30) 
At  the  maximum  value  of  C.E.M.F.  ev  =  5590  is  /  =  223.5  (31) 
At  the  maximum  value  of  current  i  —  250  is  el  =  5000  (32) 

Curve  of  zero  displacement  of  phase, 


€l  =  10  V(250  -  O2  +  4  *a  (34) 

=  10  V6.25  x  104  —  500  /  +  5  / 2 
Minimum  C.E.M.F.  point  of  this  curve, 

/  =  50         ^  =  2240  (35) 

Curve  of  maximum  displacement  of  phase, 

/  =  10  *'2  (40) 

(6.25  X  106-^2)2  +  .65  X  106  /«  -  1010/2  =  0.        (42) 


SYNCHRONOUS  MOTOR.  351 

Fig.  154  gives  the  two  ellipses  of  zero  power,  in  drawn 
lines,  with  the  curves  of  zero  displacement  in  dotted,  the 
curves  of  maximum  displacement  in  dash-dotted  lines,  and 
the  points  of  maximum  power  as  crosses. 

Fig.  155  gives  the  motor-power  characteristics,  for, 

/  =  10  kilowatts. 
p  =  50  kilowatts. 
/  =  100  kilowatts. 
p  =  150  kilowatts. 
p  =  156.25  kilowatts. 

together  with  the  curves  of  zero  displacement,  and  of  maxi- 
mum displacement. 

212.  G.    Discussion  of  Results. 

The  characteristic  curves  of  the  synchronous  motor,  as 
shown  in  Fig.  155,  have  been  observed  frequently,  with 
their  essential  features,  the  V-shaped  curve  of  no  load,  with 
the  point  rounded  off  and  the  two  legs  slightly  curved,  the 
one  concave,  the  other  convex ;  the  increased  rounding  off 
and  contraction  of  the  curves  with  increasing  load ;  and 
the  gradual  shifting  of  the  point  of  minimum  current  with 
increasing  load,  first  towards  lower,  then  towards  higher, 
values  of  C.E.M.F.  el. 

The  upper  parts  of  the  curves,  however,  I  have  never 
been  able  to  observe  experimentally,  and  consider  it  as 
probable  that  they  correspond  to  a  condition  of  synchro- 
nous motor-running,  which  is  unstable.  The  experimental 
observations  usually  extend  about  over  that  part  of  the 
curves  of  Fig.  155  which  is  reproduced  in  Fig.  156,  and  in 
trying  to  extend  the  curves  further  to  either  side,  the  motor 
is  thrown  out  of  synchronism. 

It  must  be  understood,  however,  that  these  power  char- 
acteristics of  the  synchronous  motor  in  Fig.  155  can  be  con- 
sidered as  approximations  only,  since  a  number  of  assump- 


352 


ALTERNA  TING-CURRENT  PHENOMENA. 


tions  are  made  which  are   not,  or  only  partly,  fulfilled   in 
practice.     The  foremost  of  these  are  :    • 

1.  It  is  assumed  that  el  can  be  varied  unrestrictedly, 
while  in  reality  the  possible  increase  of  el  is  limited  by 
magnetic  saturation.  Thus  in  Fig.  155,  at  an  impressed 
E.M.F.,  eQ  =  2,500  volts,  el  rises  up  to  5,590  volts,  which 
may  or  may  not  be  beyond  that  which  can  be  produced 
by  the  motor,  but  certainly  is  beyond  that  which  can  be 
constantly  given  by  the  motor. 


Fig.  156. 

2.  The  reactance,  x,  is  assumed  as  constant.  While 
the  reactance  of  the  line  is  practically  constant,  that  of  the 
motor  is  not,  but  varies  more  or  less  with  the  saturation, 
decreasing  for  higher  values.  This  decrease  of  x  increases 
the  current  /,  corresponding  to  higher  values  of  elt  and 
thereby  bends  the  curves  upwards  at  a  lower  value  of  ^ 
than  represented  in  Fig.  155. 

It  must  be  understood  that  the  motor  reactance  is  not 
a  simple  quantity,  but  represents  the  combined  effect  of 


SYNCHRONOUS  MOTOR.  353 

self-induction,  that  is,  the  E.M.F.  induced  in  the  armature 
conductor  by  the  current  flowing  therein  and  armature 
reaction,  or  the  variation  of  the  C. E.M.F.  of  the  motor 
by  the  change  of  the  resultant  field,  due  to  the  superposi- 
tion of  the  M.M.F.  of  the  armature  current  upon  the  field 
excitation ;  that  is,  it  is  the  "  synchronous  reactance." 

3.  These  curves  in  Fig.  155  represent  the  conditions 
of  constant  electric  power  of  the  motor,  thus  including  the 
mechanical  and  the  magnetic  friction  (core  loss).  While 
the  mechanical  friction  can  be  considered  as  approximately 
constant,  the  magnetic  friction  is  not,  but  increases  with 
the  magnetic  induction ;  that  is,  with  elf  and  the  same  holds 
for  the  power  consumed  for  field  excitation. 

Hence  the  useful  mechanical  output  of  the  motor  will 
on  the  same  curve,  /  =  const.,  be  larger  at  points  of  lower 
C.E.M.F.,  elt  than  at  points  of  higher  e^\  and  if  the  curves 
are  plotted  for  constant  useful  mechanical  output,  the  whole 
system  of  curves  will  be  shifted  somewhat  towards  lower 
values  of  ^ ;  hence  the  points  of  maximum  output  of  the 
motor  correspond  to  a  lower  E.M.F.  also. 

It  is  obvious  that  the  -true  mechanical  power-character- 
istics of  the  synchronous  motor  can  be  determined  only 
in  the  case  of  the  particular  conditions  of  the  installation 
under  consideration. 


354  AL  TERN  A  TING-CURRENT  PHENOMENA, 


CHAPTER    XX. 

COMMUTATOR  MOTORS. 

213.  Commutator   motors  —  that   is,   motors  in  which 
the   current   enters   or  leaves    the  armature  over  brushes 
through    a    segmental    commutator  —  have    been    built    of 
various  types,   but    have   not  found    any  extensive   appli- 
cation, in  consequence  of  the  superiority  of  the  induction 
and   synchronous  motors,  due  to  the  absence  of  commu- 
tators. 

The  main  subdivisions  of  commutator  motcrs  are  the 
repulsion  motor,  the  series  motor,  and  the  shunt  motor. 

REPULSION    MOTOR. 

214.  The   repulsion    motor  -is  an   induction  motor  or 
transformer  motor ;  that   is,  a  motor  in  which  the   main 
current   enters  the  primary  member  or  field   only,   while 
in  the   secondary  member,  or  armature,   a  current  is   in- 
duced, arid  thus  the  action  is  due  to  the  repulsive  thrust 
between  induced  current  and  inducing  magnetism. 

As  stated  under  the  heading  of  induction  motors,  a 
multiple  circuit  armature  is  required  for  the  purpose  of 
having  always  secondary  circuits  in  inductive  relation  to 
the  primary  circuit  during  the  rotation.  If  with  a  single- 
coil  field,  these  secondary  circuits  are  constantly  closed 
upon  themselves  as  in  the  induction  motor,  the  primary 
circuit  will  not  exert  a  rotary  effect  upon  the  armature 
while  at  rest,  since  in  half  of  the  armature  coils  the  cur- 
rent is  induced  so  as  to  give  a  rotary  effort  in  the  one 
direction,  and  in  the  other  half  the  current  is  induced  to 


COMMUTATOR  MOTORS. 


355 


give  a  rotary  effort   in  the   opposite  direction,   as   shown 
by  the  arrows  in  Fig.  157. 

In  the  induction  motor  a  second  magnetic  field  is  used 
to  act  upon  the  currents  induced  by  the  first,  or  inducing 
magnetic  field,  and  thereby  cause  a  rotation.  That  means 
the  motor  consists  of  a  primary  electric  circuit,  inducing 


Fig.  157. 

in  the  armature  the  secondary  currents,  and  a  primary 
magnetizing  circuit  producing  the  magnetism  to  act  upon 
the  secondary  currents. 

In  the  polyphase  induction  motor  both  functions  of  the 
primary  circuit  are  usually  combined  in  the  same  coils  ;  that 
is,  each  primary  coil  induces  secondary  currents,  and  pro- 
duces magnetic  flux  acting  upon  secondary  currents  induced 
by  another  primary  coil. 


356 


AL  TERNA  TING-CURRENT  PHENOMENA. 


215.  In  the  repulsion  motor  the  difficulty  due  to  the 
equal  and  opposite  rotary  efforts,  caused  by  the  induced 
armature  currents  when  acted  upon  by  the  inducing  mag- 
netic field,  is  overcome  by  having  the  armature  coils  closed 
upon  themselves,  either  on  short  circuit  or  through  resist- 
ance, only  in  that  position  where  the  induced  currents  give 


Fig.  158. 

a  rotary  effort  in  the  desired  direction,  while  the  armature 
coils  are  open-circuited  in  the  position  where  the  rotary 
effort  of  the  induced  currents  would  be  in  opposition  to 
the  desired  rotation.  This  requires  means  to  open  or  close 
the  circuit  of  the  armature  coils  and  thereby  introduces  the 
commutator. 

Thus  the  general  construction  of  a  repulsion  motor  is 
as  shown  in  Figs.  158  and  159  diagrammatically  as  bipolar 


COMMUTATOR  MOTORS. 


357 


motor.  The  field  is  a  single-phase  alternating  field  F,  the 
armature  shown  diagrammatically  as  ring  wound  A  consists 
of  a  number  of  coils  connected  to  a  segmental  commutator 
C,  in  general  in  the  same  way  as  in  continuous-current  ma- 
chines. Brushes  standing  under  an  angle  of  about  45°  with 
the  direction  of  the  magnetic  field,  short-circuit  either  a 


Fig.  159. 

part  of  the  armature  coils  as  shown  in  Fig.  158,  or  the 
whole  armature  by  a  connection  from  brush  to  brush  as 
shown  in  Fig.  159. 

The  former  arrangement  has  the  disadvantage  of  using  a 
part  of  the  armature  coils  only.  The  second  arrangement 
has  the  disadvantage  that,  in  the  passage  of  the  brush  from 
segment  to  segment,  individual  armature  coils  are  short- 


358 


AL  TERNA  TING-CURRENT  PHENOMENA. 


circuited,  and  thereby  give  a  torque  in  opposite  direction  to 
the  torque  developed  by  the  main  induced  current  flowing 
through  the  whole  armature  from  brush  to  brush. 

216.  Thus  the  repulsion  motor  consists  of  a  primary 
electric  circuit,  a  magnetic  circuit  interlinked  therewith, 
and  a  secondary  circuit  closed  upon  itself  and  displaced  in 


Fig.  160. 

space  by  45°  —  in  a  bipolar  motor  —  from  the  direction  of 
the  magnetic  flux,  as  shown  diagrammatically  in  Fig.  160.  * 

This  secondary  circuit,  while  set  in  motion,  still  remains 
in  the  same  position  of  45°  displacement,  with  the  magnetic 
flux,  or  rather,  what  is  theoretically  the  same,  when  moving 
out  of  this  position,  is  replaced  by  other  secondary  circuits 
entering  this  position  of  45°  displacement. 

For  simplicity,  in  the  following  all  the  secondary  quan- 


COMMUTATOR  MOTORS.  359 

titles,  as  E.M.F.,  current,  resistance,  reactance,  etc.,  are 
assumed  as  reduced  to  the  primary  circuit  by  the  ratio  of 
turns,  in  the  same  way  as  done  in  the  chapter  on  Induction 

Motors. 

217.    Let 

$  =  maximum  magnetic  flux  per  field  pole  ; 
e   =  effective  E.M.F.  induced  thereby  in  the  field  turns  ;  thus, 


where         ;/  =  number  of  turns,  N=  frequency. 

<?108 

thus,  4>  =  —  -- 

\&-anN 

The  instantaneous  value  of  magnetism  is 

<f>  =  <&  sin  (3  ; 

and  the  flux  interlinked  with  the  armature  circuit 
<£x  =  <I>  sin  /3  sin  X  ; 

when  X  is  the  angle  between  the  plane  of  the  armature  coil 
and  the  direction  of  the  magnetic  flux.     (Usually  about  45°.) 
The  E.M.F.  induced  in  the  armature  circuit,  of  n  turns, 
(as  reduced  to  primary  circuit),  is  thus, 

e  =  _  n  ^1  10-8,   =  -  n®  4-  sin  B  sin  X  lO"8, 

at  at 

=  -  n$>     sin  X  cos  (3        +  sin  (3  cos  X  10~8. 


If  N=  frequency  in  cycles  per  second,  N:  =  frequency 
of  rotation  or  speed  in  cycles  per  second,  and  k  =  N^/  N 
speed 


we  have 


frequency 


thus,    gl  =  —  2-TrnJV®  {sin  X  cos  /?  +  k  cos  X  sin  B\  10~8, 
or,  since  $  =  — — —  — , 

et  =  e  V2  {sin  X  cos  /3  +  k  cos  X  sin  fi\. 


360  ALTERNATING-CURRENT  PHENOMENA. 

218.    Introducing  now  complex  quantities,  and  counting 
the  time  from  the  zero  value  of  rising  magnetism,  the  mag- 
netism is  represented  by      /4>, 
the  primary  induced  E.M.F.,   E  =  —  e, 
the  secondary  induced  E.M.F.,   £1  =  —  e  {sin  X  +j"k  cos  X|; 
hence,  if 
Zl  =  r1—jx1=  secondary  impedance  reduced  to  primary  circuit, 

Z  =  r  —  jx  =  primary  impedance, 

Y  =  g  —jb   =  exciting  admittance, 

we  have, 

&  sin  X  -f-  jk  cos  A 

secondary  current,    7X  =  —  L  =  -  e  -    _      -  , 

primary  exciting  current,  I0  =  eY=  e  (g  +jb}, 
hence,  total  primary  current, 


Primary  impressed  E.M.F.,  E0=  —  E  +  IZ\ 
=  e     1  +  (sinX 


Neglecting  in  E0  the  last  term,  as  of  higher  order, 

£0  =  e  j  1  +  sin  X  +jk  cos  X  ^  ^4^  j  ; 
or,  eliminating  imaginary  quantities, 

e  V(?i  +  r  sin  X  -f-  kx  cos  X)2  +  (x^  +  x  sin  X  —  kr  cos  X)2 

The  power  consumed  by  the  component  of  primary 
counter  E.M.F.,  whose  flux  is  interlinked  with  the  secondary 
e  sin  X,  is, 

f  =  [e  sin  X  /]'  =  ^inXfosuiX-^cosX)  , 

r\  +  x\ 
the  power  consumed  by  the  secondary  resistance  is, 

_    2     _  **ri  (sin2  x  +  ^  cos2  x) 

hence  the  difference,  or  the  mechanical  power  developed  by 
the  motor  armature, 


COMMUTATOR   MOTORS.  361 


and  substituting  for  e, 

egk  cos  X  (x^  sin  X  +  r^k  cos  X) 

~  fa  +  r  sin  X  +  kx  cos  X)2  +  (xl  +  x  sin  \  —  kr  cos  X)2 ' 
and  the  torque  in  synchronous  watts, 

P  <?02  cos  X  (x1  sin  X  +  r^k  cos  X) 

~~  /£  ~~  (/i  +  ?"  sin  A  +  £#  cos  X)2  +  (xt  +  x  sin  X  —  kr  cos  X)2 
or         T=  V27r^lO-8  [/!<!>  sin  X  7X  cos  A]'  =  [^/!  cos  X}> 
_  ^  cos  X  (xl  sin  X  +  r^k  cos  X) 

r2  +  x2 
The  stationary  torque  is,  k  =  0, 

_  ifo2^  sin  X  cos  X 

0  =  (rx  +  r  sin  X)2  +  (^  +  *  sin  X)2 ' 

and  neglecting  the  primary  impedance,  r  =  0  =  x, 
_  e^x^  sin  X  cos  X  _  (fo2^  sin2  X 

which  is  a  maximum  at  X  =  45°. 
At  speed  k,  neglecting  r  =  0  =  x, 

<?02  cos  X  (X  sin  X  +  r^k  cos  X) 
— r2  j-^2 —          ~' 

which  is  a  maximum  for  - —  =  0,  which  gives, 

cot  2  X  =  — .     For  k  =  0,  X  =  45° ;  for  k  =  oo  ,  X  =  0. 

that  is,  in  the  repulsion  motor,  with  increasing  speed,  the 
angle  of  secondary  closed  circuit,  X,  has  to  be  reduced  to 
get  maximum  torque. 

219.    At  A  =  45°  we  have, 


(rx  V2  +  r  +  £*)2  +  (^  V2  +  x  -  krf 
and  the  power, 

p=  ^k  (x,  +  r,K)_ 

(r,  V2  +  r  +  kx)*+(xi  ^2  +  x  -  krf' 


362  ALTERNATING-CURRENT  PHENOMENA. 

this  is  a  maximum,  at  constant  X  =  45°,  for  — —  =  0,  which 

dk 


gives,  k  = 1 

At  X  =  0  we  have, 
T-- 


fa  +  kxf  +  (*t  -  krf 

that  is,  T  =  0  at  k  =  0,  or,  the  motor  is  not  self-starting, 
when  X  =  0. 


P  = 


dP 


which  is  a  maximum  at  constant  X  =  0  for,  -—  =  0,  which 

dk 
gives, 


rx-,  —  xr-. 


MOO 

-- 

'..i  i'j 

S 

•^ 

"~ 

m 

-t^> 

, 

/ 

no 

1 

/ 

/ 

R 

:PL 

LS 

ON 

M( 

5TC 

3R 

;••') 

m 

0 

/ 

V 

OC 

rt 

/ 

/ 

r= 

.! 

r,  ' 

05 

joa 

> 

/ 

X 

2. 

x. 

1. 

M 

/ 

p- 

DO 

1.17 

0  1 

j^  ^ 
<) 

g 

k 
14 

—  i, 
-,] 

I 

21  K  I 

/ 

UW 

K1 

F^ 

£d_ 

/ 

s 

2. 

I) 

F/fir.  161.    Repulsion  Motor. 

As  an  instance  is  shown,  in  Fig.  161,  the  power  output 
as  ordinates,  with  the  speed  k  =  N^_  /  N  as  abscissae,  of  a 
repulsion  motor  of  the  constants, 

X  =  45°        e0  =  100. 
r=    .1         r1=    .05 
*  =  2.0        *x  =  1.0 
giving  the  power, 

10,000  f  .02  +  1.41  k  —  .05  ffj 
~~     .171  +  2  y&)2  +  (3.14  -  .1  Kf  ' 


COMMUTATOR  MOTORS. 


SERIES    MOTOR.       SHUNT    MOTOR. 

220.  If,  in  a  continuous-current  motor,  series  motor  as 
well  as  shunt  motor,  the  current  is  reversed,  the  direction 
of  rotation  remains  the  same,  since  field  magnetism  and 
armature  current  have  reversed  their  sign,  and  their  prod- 


Fig.  162.     Series  Motor. 

net,  the  torque,  thus  maintained  the  same  sign.  There- 
fore such  a  motor,  when  supplied  by  an  alternating  current, 
will  operate  also,  provided  that  the  reversals  in  field  and 
in  armature  take  place  simultaneously.  In  the  series  motor 
this  is  necessarily  the  case,  the  same  current  passing  through 
field  and  through  armature. 

With  an  alternating  current  in  the  field,  obviously  the 


364  ALTERNATING-CURRENT  PHENOMENA. 

magnetic  circuit  has  to  be  laminated  to  exclude  eddy  cur- 
rents. 

Let,  in  a  series  'motor,  Fig.  146, 

<l>    =  effective  magnetism  per  pole, 

n     =  number  of  field  turns  per  pole  in  series, 

«i    =  number  of  armature  turns  in  series  between  brushes, 

/    =  number  of  poles, 

(R.    =  magnetic  reluctance  of  field  circuit,* 

(R!  =  magnetic  reluctance  of  armature  circuit,! 

4>i  =  effective    magnetic    flux    produced    by   armature    current 

(cross  magnetization)  per  pole, 
r     =  resistance    of    field    (effective    resistance,    including    hys- 

teresis), 
rj    =  resistance  of  armature  (effective  resistance,  including  hys- 

teresis), 

N  =  frequency  of  alternations, 
N±  =  speed  in  cycles  per  second. 

It  is  then, 

E.M.F.  induced  in    armature   conductors   by  their  rotation 
through  the  magnetic  field  (counter  E.M.F.  of  motor). 

E    =4 


E.M.F.  of  self-induction  of  field, 

E'  = 
E.M.F.  of  self-induction  of  armature, 

^/  =  27r«1^V<I>110-8, 
E.M.F.  consumed  by  resistance, 

Er  =  (r  +  *i)  I, 
where 

/  =  current  passing  through  motor,  in  amperes  effective. 

Further,  it  is  : 
Field  magnetism  :          $  =  n  7108  /  (R 

*  That  is,  the  main  magnetic  circuit  of  the  motor. 

t  That  is,  the  magnetic  circuit  of  the  cross  magnetization,  produced  by  the  armature 
reaction. 


COMMUTATOR  MOTORS.  365 


Armature  magnetism : 
Wj/108 

1  =  "V"; 

Substituting  these  values, 


(R 
ptfNI 


E'  = 

(R 

E1  =  ^^niNI . 
Er  =  (r  +  rj)  / 
Thus  the  impressed  E.M.F., 


or,  since 

i,2 


x  =  2  TT  N^-  =  reactance  of  field  ; 
(R 


2-n-jV—  =  reactance  of  armature 
fti 


and 

/ 


«  •    «, 


366  AL  TERNA  TING-CURRENT  PHENOMENA. 

221.    The  power  output  at  armature  shaft  is, 
J>=  El 


\       (R 


(R 


fi-         *Ef 

7T          «     7V^ 


/2   n±  N±  x  _j_  r  _^_ 
The  displacement  of  phase  between  current  and  E.M.F. 


tan  CD  = 


Neglecting,  as  approximation,  the  resistances  r  +  rlf  it 

1  +  |! 
lan  W  =  ?    «j  ^ 

7T  /«     7V 

^n2 


1+^' 

^ 


/«    TV 


COMMUTATOR  MOTORS.  367 

hence  a  maximum  for, 


3r 

7T 

substituting  this  in  tan  w,  it  is  : 

tan  o>  =  1,         or,         w  =  45°. 

222.    Instance  of  such  an  alternating-current  motor, 
^  =  100         AT=60         p  =  2. 

r  =  .03          ri  =  .12 
x  =  .9  *!  =  .5 

n  =  10  »j  =  48 

Special  provisions  were  made  to  keep  the  armature  re- 
actance a  minimum,  and  overcome  the  distortion  of  the 
field  by  the  armature  M.M.F.,  by  means  of  a  coil  closely 
surrounding  the  armature  and  excited  by  a  current  of  equal 
phase  but  opposite  direction  with  the  armature  current 
(Eickemeyer).  Thereby  it  was  possible  to  operate  a  two- 
circuit,  96-turn  armature  in  a  bipolar  field  of  20  turns,  at 
a  ratio  of 

armature  ampere-turns  r>  A 

field  ampere-turns 

It  is  in  this  case, 

100 


V(.023  vVi  +  ,15)2  +  1.96 

230  ./v; 

(.023  A!  +  .15)2  +  1.96 


368 


AL  TERNA  TING-CURRENT  PHENOMENA. 


In  Fig.  163  are  given,  with  the  speed  Nv  as  abscissae, 
the  values  of  current  /,  power  P,  and  power  factor  cos  o> 
of  this  motor. 


SER 

ES 

MO 

FOP 

Er 

00 

^ 

Vaf- 

3 

r  = 

n= 

03 
.12 

=(, 
x 

_.y 

=  .5 

>->w 

N  = 

60 

P= 

2 

0,, 

hi 

TUMI 

_x 

^ 

^~~~ 

^« 

< 

^ 

2Stt> 

s 

V( 

J23 

](     ~ 

QjF 

1.9 

gem 

/ 

/ 

1Z'_. 

NI 

•-il(N) 

/ 

\'(. 

[23 

^,  - 

-•)- 

'   9 

•>->00 

/ 

* 

/ 

| 

•JIHIII 

/ 

V( 

).n 

^  ^ 

'SI-' 

1.9 

M 

Am 

P- 

1S'K> 

/ 

u 

SO 

icon 

/ 

cos 

£>___ 



-sr 

_70 

ij 

•\ 

po* 

ev  ^ 

70 

H 

£ 

'  — 

— 

______ 

^J 

•< 

GO 

50 

1000 

^ 

><~~ 

•*-. 

^_<ii 

I 

40 

/ 

sew 

X 

^ 

•  . 

_Jq 

ssa 

t 

40 

30 

GOO, 

x 

;',o 

I 

>/ 

M 

g 

% 

01  111 

HI 

'oN 

no 

20 

30 

40 

50 

00 

Q 

B 

0 

Fig.  163.     Series  Motor. 

223.  The  shunt  motor  with  laminated  field  will  not 
operate  satisfactorily  in  an  alternating-current  circuit.  It 
will  start  with  good  torque,  since  in  starting  the  current  in 
armature,  as  well  as  in  field,  are  greatly  lagging,  and  thus 
approximately  in  phase  with  each  other.  With  increasing 
speed,  however,  the  armature  current  should  come  more 
into  phase  with  the  impressed  E.M.F.,  to  represent  power. 
Since,  however,  the  field  current,  and  thus  the  field  mag 
netism,  lag  nearly  90°,  the  induced  E.M.F.  of  the  armature 
rotation  will  lag  nearly  90°,  and  thus  not  represent  power. 


COMMUTATOR  MOTORS.  369 

Hence,  to  make  a  shunt  motor  work  on  alternating-cur- 
rent circuits,  the  magnetism  of  the  field  should  be  approxi- 
mately in  phase  with  the  impressed  E.M.F.,  that  is,  the  field 
reactance  negligible.  Since  the  self-induction  of  the  field  is 
far  in  excess  to  its  resistance,  this  requires  the  insertion  of 
negative  reactance,  or  capacity,  in  the  field. 

If  the  self-induction  of  the  field  circuit  is  balanced  by 
capacity,  the  motor  will  operate,  provided  that  the  armature 
reactance  is  low,  and  that  in  starting  sufficient  resistance 
is  inserted  in  the  armature  circuit  to  keep  the  armature 
current  approximately  in  phase  with  the  E.M.F.  Under 
these  conditions  the  equations  of  the  motor  will  be  similar 
to  those  of  the  series  motor. 

However,  such  motors  have  not  been  introduced,  due  to 
the  difficulty  of  maintaining  the  balance  between  capacity 
and  self-induction  in  the  field  circuit,  which  depends  upon 
the  square  of  the  frequency,  and  thus  is  disturbed  by  the 
least  change  of  frequency. 

The  main  objection  to  both  series  and  shunt  motors  is 
the  destructive  sparking  at  the  commutator  due  to  the  in- 
duction of  secondary  currents  in  those  armature  coils  which 
pass  under  the  brushes.  As  seen  in  Fig.  162,  with  the 
normal  position  of  brushes  midway  between  the  field  poles, 
the  armature  coil  which  passes  under  the  brush  incloses  the 
total  magnetic  flux.  Thus,  in  this  moment  no  E.M.F.  is 
induced  in  the  armature  coil  due  to  its  rotation,  but  the 
E.M.F.  induced  by  the  alternation  of  the  magnetic  flux 
has  a  maximum  at  this  moment,  and  the  coil,  when  short- 
circuited  by  the  brush,  acts  as  a  short-circuited  secondary 
to  the  field  coils  as  primary  ;  that  is,  an  excessive  current 
flows  through  this  armature  coil,  which  either  destroys  it, 
or  at  least  causes  vicious  sparking  when  interrupted  by  the 
motion  of  the  arm'ature. 

To  overcome  this  difficulty  various  arrangements  have 
been  proposed,  but  have  not  found  an  application. 


370  ALTERNATING-CURRENT  PHENOMENA. 

224.  Compared  with  the  synchronous  motor  which  has 
practically  no  lagging  currents,  and  the  induction  motor 
which  reaches  very  high  power  factors,  the  power  factor  of 
the  series  motor  is  low,  as  seen  from  Fig.  163,  which  repre- 
sents about  the  best  possible  design  of  such  motors. 

In  the  alternating-series  motor,  as  well  as  in  the  shunt 
motor,  no  position  of  an  armature  coil  exists  wherein  the 
coil  is  dead;  but  in  every  position  E.M.F.  is  induced  in  the 
armature  coil :  in  the  position  parallel  with  the  field  flux  an 
E.M.F.  in  phase  with  the  current,  in  the  position  at  right 
angles  with  the  field  flux  an  E.M.F.  in  quadrature  with  the 
current,  intermediate  E.M.Fs.  in  intermediate  positions. 
At  the  speed  irJV/2  the  two  induced  E.M.Fs.  in  phase  and 
in  quadrature  with  the  current  are  equal,  and  the  armature 
coils  are  the  seat  of  a  complete  system  of  symmetrical  and 
balanced  polyphase  E.M.Fs.  Thus,  by  means  of  stationary 
brushes,  from  such  a  commutator  polyphase  currents  could 
be  derived. 


REACTION  MACHINES.  371 


CHAPTER    XXI. 

REACTION  MACHINES. 

225.  In  the  chapters  on  Alternating-Current  Genera- 
tors and  on  Induction  Motors,  the  assumption  has  been 
made  that  the  reactance  x  of  the  machine  is  a  constant. 
While  this  is  more  or  less  approximately  the  case  in  many 
alternators,  in  others,  especially  in  machines  of  large  arma- 
ture reaction,  the  reactance  x  is  variable,  and  is  different  in 
the  different  positions  of  the  armature  coils  in  the  magnetic 
circuit.  This  variation  of  the  reactance  causes  phenomena 
which  do  not  find  their  explanation  by  the  theoretical  cal- 
culations made  under  the  assumption  of  constant  reactance. 

It  is  known  that  synchronous  motors  of  large  and 
variable  reactance  keep  in  synchronism,  and  are  able  to 
do  a  considerable  amount  of  work,  and  even  carry  under 
circumstances  full  load,  if  the  field-exciting  circuit  is 
broken,  and  thereby  the  counter  E.M.F.  E±  reduced  to 
zero,  and  sometimes  even  if  the  field  circuit  is  reversed 
and  the  counter  E.M.F.  E±  made  negative. 

Inversely,  under  certain  conditions  of  load,  the  current 
and  the  E.M.F.  of  a  generator  do  not  disappear  if  the  gene- 
rator field  is  broken,  or  even  reversed  to  a  small  negative 
value,  in  which  latter  case  the  current  flows  against  the 
E.M.F.  EQ  of  the  generator. 

Furthermore,  a  shuttle  armature  without  any  winding 
will  in  an  alternating  magnetic  field  revolve  when  once 
brought  up  to  synchronism,  and  do  considerable  work  as 
a  motor. 

These  phenomena  are  not  due  to  remanent  magnetism 
nor  to  the  magnetizing  effect  of  Foucault  currents,  because 


372  AL  TERNA  TING-CURRENT  PHENOMENA. 

they  exist  also  in  machines  with  laminated  fields,  and  exist 
if  the  alternator  is  brought  up  to  synchronism  by  external 
means  and  the  remanent  magnetism  of  the  field  poles  de- 
stroyed beforehand  by  application  of  an  alternating  current. 

226.  These  phenomena  cannot  be  explained  under  the 
assumption  of  a  constant  synchronous  reactance;  because 
in  this  case,  at  no-field  excitation,  the  E.M.F.  or  counter 
E.M.F.  of  the  machine  is  zero,  and  the  only  E.M.F.  exist- 
ing in  the  alternator  is  the  E.M.F.  of  self-induction;  that 
is,   the    E.M.F.   induced  by  the  alternating  current  upon 
itself.     If,  however,  the  synchronous  reactance  is  constant, 
the  counter  E.M.F.  of  self-induction  is  in  quadrature  with 
the  current  and  wattless;  that  is,  can  neither  produce  nor 
consume  energy. 

'  In  the  synchronous  motor  running  without  field  excita- 
tion, always  a  large  lag  of  the  current  behind  the  impressed 
E.M.F.  exists;  and  an  alternating  generator  will  yield  an 
E.M.F.  without  field  excitation,  only  when  closed  by  an 
external  circuit  of  large  negative  reactance ;  that  is,  a  circuit 
in  which  the  current  leads  the  E.M.F.,  as  a  condenser,  or 
an  over-excited  synchronous  motor,  etc. 

Self-excitation  of  the  alternator  by  armature  reaction 
can  be  explained  by  the  fact  that  the  counter  E.M.F.  of 
self-induction  is  not  wattless  or  in  quadrature  with  the  cur- 
rent, but  contains  an  energy  component ;  that  is,  that  the 
reactance  is  of  the  form  X  =  h  —jx,  where  x  is  the  wattless 
component  of  reactance  and  h  the  energy  component  of 
reactance,  and  h  is  positive  if  the  reactance  consumes 
power,  —  in  which  case  the  counter  E.M.F.  of  self-induc- 
tion lags  more  than  90°  behind  the  current, — while  h  is 
negative  if  the  reactance  produces  power,  —  in  which  case 
the  counter  E.M.F.  of  self-induction  lags  less  than  90° 
behind  the  current. 

227.  A  case  of  this  nature  has  been  discussed  already 
in  the  chapter  on  Hysteresis,  from  a  different  point  of  view. 


REACTION  MACHINES.  373 

There  the  effect  of  magnetic  hysteresis  was  found  to  distort 
the  current  wave  in  such  a  way  that  the  equivalent  sine 
wave,  that  is,  the  sine  wave  of  equal  effective  strength  and 
equal  power  with  the  distorted  wave,  is  in  advance  of  the 
wave  of  magnetism  by  what  is  called  the  angle  of  hysteretic 
advance  of  phase  a.  Since  the  E.M.F.  induced  by  the 
magnetism,  or  counter  E.M.F.  of  self-induction,  lags  90° 
behind  the  magnetism,  it  lags  90  -f-  a  behind  the  current ; 
that  is,  the  self-induction  in  a  circuit  containing  iron  is  not 
in  quadrature  with  the  current  and  thereby  wattless,  but 
lags  more  than  90°  and  thereby  consumes  power,  so  that 
the  reactance  has  to  be  represented  by  X  =  Ji  —jx,  where 
h  is  what  has  been  called  the  "  effective  hysteretic  resis- 
tance." 

A  similar  phenomenon  takes  place  in  alternators  of  vari- 
able reactance,  or  what  is  the  same,  variable  magnetic 
reluctance. 

228.  Obviously,  if  the  reactance  or  reluctance  is  vari- 
able, it  will  perform  a  complete  cycle  during  the  time  the 
armature  coil  moves  from  one  field  pole  to  the  next  field 
pole,  that  is,  during  one-half  wave  of  the  main  current. 
That  is,  in  other  words,  the  reluctance  and  reactance  vary 
with  twice  the  frequency  of  the  alternating  main  current. 
Such  a  case  is  shown  in  Figs..  164  and  165.  The  impressed 
E.M.F.,  and  thus  at  negligible  resistance,  the  counter  E.M.F., 
is  represented  by  the  sine  wave  E,  thus  the  magnetism  pro- 
duced thereby  is  a  sine  wave  4>,  90°  ahead  of  E.  The 
reactance  is  represented  by  the  sine  wave  x,  varying  with 
the  double  frequency  of  E,  and  shown  in  Fig.  164  to  reach 
the  maximum  value  during  the  rise  of  magnetism,  in  Fig. 
165  during  the  decrease  of  magnetism.  The  current  /  re- 
quired to  produce  the  magnetism  <l>  is  found  from  3>  and-^r 
in  combination  with  the  cycle  of  molecular  magnetic  friction 
of  the  material,  and  the  power  P  is  the  product  IE  As 
seen  in  Fig.  164,  the  positive  part  of  P  is  larger  than  the 


374  AL  TERNA  TING-CURRENT  PHENOMENA. 


f, 

^ 

/' 

\ 

<p 

/ 

\ 

^ 

^ 

^  — 

> 

^ 

E 

X 

/ 

/ 

i 

/ 

s 

\ 

/ 

\ 

/ 

/ 

i, 

A 

\ 

s 

V 

\ 

2 

^ 

s~~~ 

\^ 

// 

"\ 

s 

* 

^ 

\ 

•^ 

\ 

> 

. 

}, 

•^  / 

s 

y 

^ 

/ 

\    ^ 

\ 

\ 

1 

— 

\ 

// 

i 

1  — 

\ 

\ 

\/ 

\\ 

/ 

\ 

i 

\ 

y 

\\ 

A 

Vs 

•* 

\ 

I 

'^^ 

^ 

/ 

V 

_^- 

—  ' 

\ 

\ 

k*s' 

x^^ 

\ 

I 

/ 

\ 

^ 

s. 

\ 

N 

\ 

y 

I 

/ 

\ 

\ 

k 

\ 

\ 

/ 

r\ 

S 

\ 

\ 

N 

^_ 

\ 

x 

b:S 

\ 

\ 

I 

V 

9 

Fig.   164,     Variable  Reactance,  Reaction  Machine. 


Fig.  165.     Variable  Reactance,  Reaction  Machine. 


REACTION  MACHINES. 


375 


negative  part ;  that  is,  the  machine  produces  electrical  energy 
as  generator.  In  Fig.  165  the  negative  part  of  P  is  larger 
than  the  positive ;  that  is,  the  machine  consumes  electrical 
energy  and  produces  mechanical  energy  as  synchronous 
mqtor.  In  Figs.  166  and  167  are  given  the  two  hysteretic 
cycles  or  looped  curves  <J>,  /  under  the  two  conditions.  They 
show  that,  due  to  the  variation  of  reactance  x,  in  the  first 
case  the  hysteretic  cycle  has  been  overturned  so  as  to 
represent  not  consumption,  but  production  of  electrical 


- 


Fig.   166.     Hysteretic  Loop  of  Reaction  Machine. 


energy,  while  in  the  second  case  the  hysteretic  cycle  has 
been  widened,  representing  not  only  the  electrical  energy 
consumed  by  molecular  magnetic  friction,  but  also  the  me- 
chanical output. 

229.  It  is  evident  that  the  variation  of  reluctance  must 
be  symmetrical  with  regard  to  the  field  poles  ;  that  is,  that 
the  two  extreme  values  of  reluctance,  maximum  and  mini- 
mum, will  take  place  at  the  moment  where  the  armature 


J76 


ALTERNA TING-CURRENT  PHENOMENA. 


coil  stands  in  front  of  the  field  pole,  and  at  the  moment 
where  it  stands  midway  between  the  field  poles. 

The  effect  of  this  periodic  variation  of  reluctance  is  a 
distortion  of  the  wave  of  E.M.F.,  or  of  the  wave  of  current, 
or  of  both.  Here  again,  as  before,  the  distorted  wave  can 
be  replaced  by  the  equivalent  sine  wave,  or  sine  wave  of 
equal  effective  intensity  and  equal  power. 

The  instantaneous  value  of  magnetism  produced  by  the 


Fig.  167.    Hysteretic  Loop  of  Reaction  Machine. 


armature  current  —  which  magnetism  induces  in  the  arma- 
ture conductor  the  E.M.F.  of  self-induction  —  is  propor- 
tional to  the  instantaneous  value  of  the  current,  divided 
by  the  instantaneous  value  of  the  reluctance.  Since  the 
extreme  values  of  the  reluctance  coincide  with  the  sym- 
metrical positions  of  the  armature  with  regard  to  the  field 
poles,  —  that  is,  with  zero  and  maximum  value  of  the  in- 
duced E.M.F.,  EQ,  of  the  machine,  —  it  follows  that,  if  the 
current  is  in  phase  or  in  quadrature  with  the  E.M.F.  EQ, 
the  reluctance  wave  is  symmetrical  to  the  current  wave, 
and  the  wave  of  magnetism  therefore  symmetrical  to  the 


REACTION  MACHINES.  377 

current  wave  also.  Hence  the  equivalent  sine  wave  of 
magnetism  is  of  equal  phase  with  the  current  wave ;  that 
is,  the  E.M.F.  of  self-induction  lags  90°  behind  the  cur- 
rent, or  is  wattless. 

Thus  at  no-phase  displacement,  and  at  90°  phase  dis- 
placement, a  reaction  machine  can  neither  produce  electri- 
cal power  nor  mechanical  power. 

230.  If,  however,  the  current  wave  differs  in  phase 
from  the  wave  of  E.M.F.  by  less  than  90°,  but  more  than 
zero  degrees,  it  is  unsymmetrical  with  regard  to  the 
reluctance  wave,  and  the  reluctance  will  be  higher  for  ris- 
ing current  than  for  decreasing  current,  or  it  will  be 
higher  for  decreasing  than  for  rising  current,  according 
to  the  phase  relation  of  current  with  regard  to  induced 
E.M.F.,  £Q. 

In  the  first  case,  if  the  reluctance  is  higher  for  rising, 
lower  for  decreasing,  current,  the  magnetism,  which  is  pro- 
portional to  current  divided  by  reluctance,  is  higher  for 
decreasing  than  for  rising  current ;  that  is,  its  equivalent 
sine  wave  lags  behind  the  sine  wave  of  current,  and  the 
E.M.F.  or  self-induction  will  lag  more  than  90°  behind  the 
current ;  that  is,  it  will  consume  electrical  power,  and 
thereby  deliver  mechanical  power,  and  do  work  as  syn- 
chronous motor. 

In  the  second  case,  if  the  reluctance  is  lower  for  rising, 
and  higher  for  decreasing,  current,  the  magnetism  is  higher 
for  rising  than  for  decreasing  current,  or  the  equivalent  sine 
wave  of  magnetism  leads  the  sine  wave  of  the  current,  and 
the  counter  E.M.F.  at  self-induction  lags  less  than  90°  be- 
hind the  current ;  that  is,  yields  electric  power  as  generator, 
and  thereby  consumes  mechanical  power. 

In  the  first  case  the  reactance  will  be  represented  by 
X  =  h  —  jx,  similar  as  in  the  case  of  hysteresis ;  while  in 
the  second  case  the  reactance  will  be  represented  by 
X  =  -  h-  jx. 


378  ALTERNATING-CURRENT  PHENOMENA. 

231.  The  influence  of  the  periodical  variation  of  reac- 
tance will  obviously  depend  upon  the  nature  of  the  variation, 
that  is,  upon  the  shape  of  the  reactance  curve.  Since, 
however,  no  matter  what  shape  the  wave  has,  it  can  always 
be  dissolved  in  a  series  of  sine  waves  of  double  frequency, 
and  its  higher  harmonics,  in  first  approximation  the  assump- 
tion can  be  made  that  the  reactance  or  the  reluctance  vary 
with  double  frequency  of  the  main  current  ;  that  is,  are 
represented  in  the  form, 

x  =  a  +  b  cos  2  /8. 

Let  the  inductance,  or  the  coefficient  of  self-induction, 
be  represented  by  — 

L  =  I  +  <£  cos  2  /3 

=  /(I  +  y  COS  2  0) 

where         y  =  amplitude  of  variation  of  inductance. 

Let 

u>  =  angle  of  lag  of  zero  value  of  current  behind  maximum  value 
of  inductance  L. 

It  is  then,  assuming  the  current  as  sine  wave,  or  repla- 
cing it  by  the  equivalent  sine  wave  of  effective  intensity  /, 

Current, 

*  =  I V2  sin  (/?  -  £). 

The  magnetism  produced  by  this  current  is, 


where  n  =  number  of  turns. 
Hence,  substituted, 


sin  (/?  -  5)  (1  +  y  cos  2  0), 
or,  expanded, 


n 
when  neglecting  the  term  of  triple  frequency,  as  wattless. 


REACTION  MACHINES,  379 

Thus  the  E.M.F.  induced  by  this  magnetism  is, 


hence,  expanded  — 

e  =  -  2  TT  7W7  V2  !7  1  -  2\  cos  £  cos  /3  +  /I  +        sn     sn 

IV      ZJ  \      2 

and  the  effective  value  of  E.M.F., 


l  +  2 


=  2  TT  NII\\  +       -  7  cos  2  a.        ^ 
Hence,  the  apparent  power,  or  the  voltamperes  — 


+  -J2  —  y  COS  2  u> 

The  instantaneous  value  of  power  is 


2sin(/?  —  c(,)f/l  —  |\  cos  w  cos  y3 + 


sin  eo  sin  /3  [.  . 
7 

and,  expanded  — 


sin  2  eo  cos2  /3  +  sin  2  /3  (  cos  2  w  —  2 \  1 
V  2/J 

Integrated,  the  effective  value  of  power  is 


380  AL  TERNA  TING-CURRENT  PHENOMENA. 

hence,  negative,  that  is,  the  machine  consumes  electrical, 
and  produces  mechanical,  power,  as  synchronous  motor,  if 
o>  >  0  ;  that  is,  with  lagging  current;  positive,  that  is,  the 
machine  produces  electrical,  and  consumes  mechanical, 
power,  as  generator,  if  to  >  0  ;  that  is,  with  leading  current. 
The  power  factor  is 

r  j_  P_  _  y  sin  2  ai 


hence,  a  maximum,  if, 

d< 

or,  expanded,  1 

cos2£  =  i 

The  power,  P,  is  a  maximum  at  given  current,  /,  if 

sin  2  w  =  1  ; 
that  is, 

to  =  45° 

at  given  E.M.F.,  E,  the  power  is 
p=  __ 


hence,  a  maximum  at 
or,  expanded, 


1  +  1T 


232.    We  have  thus,  at  impressed  E.M.F.,  E,  and  negli- 
gible resistance,  if  we  denote  the  mean  value  of  reactance, 

x=lTtNl. 
Current 


REACTION  MACHINES.  381 

Voltamperes, 

k- 


Power, 

^g2  y  sin  2  £ 


2^fl+^--ycos2 


Power  factor, 

,.              /  77    T-N                      y  sin  2  to 
f  =  cos  (E,  /)  = '  

2  y/l  +  J^  _  y  cos  2  A 
Maximum  power  at 


*+i 


Maximum  power  factor  at 


to  >  0  :  synchronous  motor,  with  lagging  current, 
w  <  0  :  generator,  with  leading  current. 

As  an  instance  is  shown  in  Fig.  168,  with  angle  to  as 
abscissae,  the  values  of  current,  power,  and  power  factor, 
for  the  constants,  — 

E  =  110 

x  =  3 

y    =.8 

hence,  j 41 

Vl.45  —  cos  2  £ 
-  2017  sin  2w 


P  = 


f=  cos  (E,I) 


1.45  —  cos  2  w 

.447  sin  2  G> 


As  seen  from  Fig.  152,  the  power  factor  /  of  such  a 
machine  is  very  low  —  does  not  exceed  40  per  cent  in  this 
instance. 


382 


ALTERNA TING-CURRENT  PHENOMENA. 


Fig.  188.    Reaction  Machine. 


DISTORTION  OF   WAVE-SHAPE.  383 


CHAPTER    XXII. 

DISTORTION   OF    WAVE-SHAPE    AND    ITS    CAUSES. 

233.  In    the   preceding   chapters   we   have    considered 
the   alternating  currents  and   alternating   E.M.Fs.  as  sine 
waves  or  as  replaced  by  their  equivalent  sine  waves. 

While  this  is  sufficiently  exact  in  most  cases,  under 
certain  circumstances  the  deviation  of  the  wave  from  sine 
shape  becomes  of  importance,  and  with  certain  distortions 
it  may  not  be  possible  to  replace  the  distorted  wave  by  an 
equivalent  sine  wave,  since  the  angle  of  phase  displacement 
of  the  equivalent  sine  wave  becomes  indefinite.  Thus  it 
becomes  desirable  to  investigate  the  distortion  of  the  wave, 
its  causes  and  its  effects. 

Since,  as  stated  before,  any  alternating  wave  can  be 
represented  by  a  series  of  sine  functions  of  odd  orders,  the 
investigation  of  distortion  of  wave-shape  resolves  itself  in 
the  investigation  of  the  higher  harmonics  of  the  alternating 
wave. 

In  general  we  have  to  distinguish  between  higher  har- 
monics of  E.M.F.  and  higher  harmonics  of  current.  Both 
depend  upon  each  other  in  so  far  as  with  a  sine  wave  of 
impressed  E.M.F.  a  distorting  effect  will  cause  distortion 
of  the  current  wave,  while  with  a  sine  wave  of  current 
passing  through  the  circuit,  a  distorting  effect  will  cause 
higher  harmonics  of  E.M.F. 

234.  In   a  conductor  revolving  with   uniform  velocity 
through  a  uniform  and  constant  magnetic  field,  a  sine  wave 
of  E.M.F.  is  induced.     In  a  circuit  with  constant  resistance 
and  constant  reactance,  this  sine  wave  of  E.M.F.  produces 


384  ALTERNATING-CURRENT  PHENOMENA. 

a  sine  wave  of  current.  Thus  distortion  of  the  wave-shape 
or  higher  harmonics  may  be  due  to  :  lack  of  uniformity  of 
the  velocity  of  the  revolving  conductor ;  lack  of  uniformity 
or  pulsation  of  the  magnetic  field  ;  pulsation  of  the  resis- 
tance ;  or  pulsation  of  the  reactance. 

The  first  two  cases,  lack  of  uniformity  of  the  rotation  or 
of  the  magnetic  field,  cause  higher  harmonics  of  E.M.F.  at 
open  circuit.  The  last,  pulsation  of  resistance  and  reac- 
tance, causes  higher  harmonics  only  with  a  current  flowing 
in  the  circuit,  that  is,  under  load. 

Lack  of  uniformity  of  the  rotation  is  of  no  practical  in- 
terest as  cause  of  distortion,  since  in  alternators,  due  to 
mechanical  momentum,  the  speed  is  always  very  nearly 
uniform  during  the  period. 

Thus  as  causes  of  higher  harmonics  remain : 

1st.  Lack  of  uniformity  and  pulsation  of  the  magnetic 
field,  causing  a  distortion  of  the  induced  E.M.F.  at  open 
circuit  as  well  as  under  load. 

2d.  Pulsation  of  the  reactance,  causing  higher  harmonics 
under  load. 

3d.  Pulsation  of  the  resistance,  causing  higher  harmonics 
under  load  also. 

Taking  up  the  different  causes  of  higher  harmonics  we 
have  :  — 

Lack  of  Uniformity  and  Pulsation  of  tJie  Magnetic  Field. 

235.  Since  most  of  the  alternating-current  generators 
contain  definite  and  sharply  defined  field  poles  covering  in 
different  types  different  proportions  of  the  pitch,  in  general 
the  magnetic  flux  interlinked  with  the  armature  coil  will 
not  vary  as  simply  sine  wave,  of  the  form  : 

$  cos  /?, 

but  as  a  complex  harmonic  function,  depending  on  the  shape 
and  the  pitch  of  the  field  poles,  and  the  arrangement  of  the 
armature  conductors.  In  this  case,  the  magnetic  flux  issu- 


DISTORTION  OF   WAVE-SHAPE.  385 

ing  from  the  field  pole  of  the  alternator  can  be  represented 
by  the  general  equation, 

4>  =  A0  +  A,  cos  /8  +  A*  cos  2(3  +  Az  cos  3/8  +  .  .  . 
+  ^  sin  £  +  -#2  sin  2  0  +  .#,  sin  3  ft  +  .  .  . 

If  the  reluctance  of  the  armature  is  uniform  in  all  directions, 
so  that  the  distribution  of  the  magnetic  flux  at  the  field-pole 
face  does  not  change  by  the  rotation  of  the  armature,  the 
rate  of  cutting  magnetic  flux  by  an  armature  conductor  is  <£, 
and  the  E.M.F.  induced  in  the  conductor  thus  equal  thereto 
in  wave  shape.  As  a  rule  A0,  Az,  At  .  .  .  By  B±  equal  zero  ; 
that  is,  successive  field  poles  are  equal  in  strength  and  dis- 
tribution of  magnetism,  but  of  opposite  polarity.  In  some 
types  of  machines,  however,  especially  induction  alternators, 
this  is  not  the  case. 

The  E.M.F.  induced  in  a  full-pitch  armature  turn  —  that 
is,  armature  conductor  and  return  conductor  distant  from 
former  by  the  pitch  of  the  armature  pole  (corresponding  to 
the  distance  from  field  pole  center  to  pole  center)  is, 
8  =  $0  -  3>180 

=  2  \Ai  cos  /3  +  Aa  cos  3  (3  +  A6  cos  5  0  +  .  .  . 
+  BI  sin  j3  +  Bz  sin  3  ft  +  jB6  sin  5  ft  +  .  .  .   \ 

Even  with  an  unsymmetrical  distribution  of  the  magnetic 
flux  in  the  air-gap,  the  E.M.F.  wave  induced  in  a  full-pitch 
armature  coil  is  symmetrical  ;  the  positive  and  negative  half 
waves  equal,  and  correspond  to  the  mean  flux  distribution 
of  adjacent  poles.  With  fractional  pitch  windings  —  that 
is,  windings  whose  turns  cover  less  than  the  armature  pole 
pitch  —  the  induced  E.M.F.  can  be  unsymmetrical  with 
unsymmetrical  magnetic  field,  but  as  a  rule  is  symmetrical 
also.  In  unitooth  alternators  the  total  induced  E.M.F.  has 
the  same  shape  as  that  induced  in  a  single  turn. 

With  the  conductors  more  or  less  distributed  over  the 
surface  of  the  armature,  the  total  induced  E.M.F.  is  the 
resultant  of  several  E.M.Fs.  of  different  phases,  and  is  thus 
more  uniformly  varying  ;  that  is,  more  sinusoidal,  approaching 


386  ALTERNATING-CURRENT  PHENOMENA. 

sine  shape,  to  within  3%  or  less,  as  for  instance  the  curves 
Fig.  169  and  Fig.  170  show,  which  represent  the  no-load 
and  full-load  wave  of  E.M.F.  of  a  three-phase  multitooth 
alternator.  The  principal  term  of  these  harmonics  is  the 
third  harmonic,  which  consequently  appears  more  or  less  in 
all  alternator  waves.  As  a  rule  these  harmonics  can  be 
considered  together  with  the  harmonics  due  to  the  varying 
reluctance  of  the  magnetic  circuit.  In  ironclad  alternators 
with  few  slots  and  teeth  per  pole,  the  passage  of  slots  across 
the  field  poles  causes  a  pulsation  of  the  magnetic  reluc- 
tance, or  its  reciprocal,  the  magnetic  inductance  of  the 
circuit.  In  consequence  thereof  the  magnetism  per  field 
pole,  or  at  least  that  part  of  the  magnetism  passing  through 
the  armature,  will  pulsate  with  a  frequency  2  y  if  y  =  num- 
ber of  slots  per  pole. 

Thus,  in  a  machine  with  one  slot  per  pole,  the  instanta- 
neous magnetic  flux  interlinked  with  the  armature  con- 
ductors can  be  expressed  by  the  equation  : 

<£  =  $  cos  /?  [1  +  e  cos  [2  (3  —  o>]  j 
where,  ®  =  average  magnetic  flux, 

c  =  amplitude  of  pulsation, 
and  to  =  phase  of  pulsation. 

In  a  machine  with  y  slots  per  pole,  the  instantaneous  flux 
interlinked  with  the  armature  conductors  will  be  : 

<f>  =  &  cos  /8  { 1  +  c  cos  [2  y  ft  —  o>]  | , 

if  the  assumption  is  made  that  the  pulsation  of  the  magnetic 
flux  follows  a  simple  sine  law,  as  first  approximation. 

In  general  the  instantaneous  magnetic  flux  interlinked 
with  the  armature  conductors  will  be  : 

^  =  *  cos  0  {1  +  6!  cos  (2  0  -  SO  +  e,  cos  (4  £  -  oV,)  +  .  .  .  f , 
where  the  term  ey  is  predominating  if  y  =  number  of  arma- 
ture slots  per  pole.  This  general  equation  includes  also  the 
effect  of  lack  of  uniformity  of  the  magnetic  flux. 


DISTORTION  OF   WAVE-SHAPE. 


387 


Nil  LoLd 


,"14   .5     y, 


Fig.  169.      No-load 


of  E.M.F.  of  Multitooth  Three-phaser. 


130 

JMtfl  I 

oad 

120      '' 

=  12 

7.0 

»= 

3  ; 

^ 

'-- 

--- 

>s, 

110 

j^ 

5 

100 

/ 

\ 

90 

j 

7 

V 

SO 

/ 

s 

70 

/ 

s 

60 

/ 

^ 

50 

/ 

\, 

10 

// 

'^ 

, 

30 

/' 

\ 

20 

/ 

\ 

10 

// 

\\ 

0 

•'/ 

/- 

"--v^ 

r- 

1  —  ^.^ 

^ 

—  V 

10 

f     ' 

10 

50 

30 

10 

g 

(50 

70 

SO 

90 

100 

no 

120 

13(1 

140 

150 

100 

170 

ISO 

Fig.  170.     Full-Load  Waue  of  E.M.F.  of  Multitooth  Three-phaser. 


388  ALTERNATING-CURRENT  PHENOMENA. 

In  case  of  a  pulsation  of  the  magnetic  flux  with  the 
frequency  2y,  due  to  an  existence  of  y  slots  per  pole  in  the 
armature,  the  instantaneous  value  of  magnetism  interlinked 
with  the  armature  coil  is  : 

<£  =  $  COS  ft  {1  +  e  COS  [2  y  ft  —  £]}. 

Hence  the  E.M.F.  induced  thereby  : 

e  =  —  n  — — 
dt 

d 

*» 

And,  expanded  : 

e=  V27rA^<fc{sin/?+e-^=—  sin[(2y  —  1)  0  -  «J] 


Hence,  the  pulsation  of  the  magnetic  flux  with  the 
frequency  2  y,  as  due  to  the  existence  of  y  slots  per  pole, 
introduces  two  harmonics,  of  the  orders  (2  y  —  1)  and 
(2  7+1). 

236.    If  y  =  1  it  is  : 
e  =  V2  TT  Nn  <i>  (sin  /3  +  1  sin  (0  —  £)  +  ^  sin  (3  /?  -  £)} ; 

that  is :  In  a  unitooth  single-phaser  a  pronounced  triple 
harmonic  may  be  expected,  but  no  pronounced  higher 
harmonics. 

Fig.  171  shows  the  wave  of  E.M.F.  of  the  main  coil  of 
a  monocyclic  alternator  at  no  load,  represented  by : 

e  =  E  (sin  (3  —  .242  sin  (  3  /3  —  6.3)  —  .046  sin  (5/3-  2.6) 
+  .068  sin  (7  £  —  3.3)  —  .027  sin  (9  ft  —  10.0)  —  .018  sin 
(11  /3  -  6.6)  +  .029  sin  (13  ft  -  8.2)}; 

hence  giving  a  pronounced  triple  harmonic  only,  as  expected. 
If  y  =  2,  it  is  : 

e  =  V2  TT  Nn  4>  j  sin  £  +  ^  sin  (3  ft  -  «J)  +  |f  sin  (5  ft  -  Si) 


DISTORTION  OF   WAVE-SHAPE. 


389 


the  no-load  wave  of  a  unitooth  quarter-phase  machine,  hav- 
ing pronounced  triple  and  quintuple  harmonics. 
If  7  =  3,  it  is  : 


in/3+       sin(5j8—  fi)  +       sin  (7  ft  -  S>)  I  . 


That  is  :  In  a  unitooth  three-phaser,  a  pronounced  quin- 
tuple and  septuple  harmonic  may  be  expected,  but  no  pro- 
nounced triple  harmonic. 


Fig.  155.     No-load  Wave  of  E.M.F.  of  Unitooth  Monocyclic  Alternator. 


Fig.  156  shows  the  wave  of  E.M.F.  of  a  unitooth  three- 
phaser  at  no  load,  represented  by : 

e  =  E  (sin  /3  —  .12  sin  (3  £  —  2.3)  —  .23  sin  (5  (3  —  1.5)  +  .134  sin 
(7  ft  _  6.2)  -  .002  sin  (9  /3  +  27.7)  -  .046  sin  (11  /?  — 
5.5)  +.031  sin  (13)8-61.5)}. 

Thus  giving  a  pronounced  quintuple  and  septuple  and 
a  lesser  triple  harmonic,  probably  due  to  the  deviation  of, 
the  field  from  uniformity,  as  explained  above,  and  deviation 
of  the  pulsation  of  reluctance  from  sine  shape.  In  some 
especially  favorable  cases,  harmonics  as  high  as  the  23d  and 
25th  have  been  observed,  caused  by  pulsation  of  the  reluc- 
tance. 


390  ALTERNATING-CURRENT  PHENOMENA. 


V 


100 


50       60       70     80        90       1 00 


30   140   150    160  170    180 


Fig.  172.    No-load  Wave  of  E.M.F.  of  Unitooth  Three-phase  Alternator. 


In  general,  if  the  pulsation  of  the  magnetic  inductance 
is  denoted  by  the  general  expression : 

l  +  ^"cYcos(2yj8-aY), 
1 

the  instantaneous  magnetic  flux  is  : 

00 


=  $  cos  13 


ey  cos  (2  y  ff  - 


cos((2y+l) 


hence,  the  E.M.F. 


2 ;  sm(P  — 


DISTORTION  OF   WAVE-SHAPE.  391 

Pulsation  of  Reactance. 

237.  The  main  causes  of  a  pulsation  of  reactance  are : 
magnetic  saturation  and  hysteresis,  and  synchronous  motion. 
Since  in  an  ironclad  magnetic  circuit  the  magnetism  is  not 
proportional  to  the   M.M.F.,  the  wave  of  magnetism  and 
thus  the  wave  of  E.M.F.  will  differ  from  the  wave  of  cur- 
rent.    As  far  as  this  distortion  is  due  to  the  variation  of 
permeability,  the   distortion   is   symmetrical  and  the  wave 
of  induced  E.M.F. 'represents  no   power.      The  distortion 
caused  by  hysteresis,  or  the  lag  of  the  magnetism  behind 
the  M.M.F.,  causes  an  unsymmetrical  distortion  of  the  wave 
which  makes  the  wave  of  induced  E.M.F.  differ  by  more 
than   90°  from  the   current  wave  and  thereby  represents 
power,  —  the  power  consumed  by  hysteresis. 

In  practice  both  effects  are  always  superimposed ;  that 
is,  in  a  ferric  inductance,  a  distortion  of  wave-shape  takes 
place  due  to  the  lack  of  proportionality  between  magnetism 
and  M.M.F.  as  expressed  by  the  variation  in  the  hysteretic 
cycle. 

This  pulsation  of  reactance  gives  rise  to  a  distortion 
consisting  mainly  of  a  triple  harmonic.  Such  current  waves 
distorted  by  hysteresis,  with  a  sine  wave  of  impressed 
E.M.F.,  are  shown  in  Figs.  66  to  69,  Chapter  X.,  on  Hy- 
steresis. Inversely,  if  the  current  is  a  sine  wave,  the  mag- 
netism and  the  E.M.F.  will  differ  from  sine  shape. 

For  further  discussion  of  this  distortion  of  wave-shape 
by  hysteresis,  Chapter  X.  may  be  consulted. 

238.  Distortion  of  wave-shape  takes  place  also  by  the 
pulsation  of  reactance  due  to  synchronous  rotation,  as  dis- 
cussed in  chapter  on  Reaction  Machines. 

In  Figs.   148   and  149,   at   a  sine  wave    of    impressed 
E.M.F.,  the  distorted  current  waves  have  been  constructed. 
Inversely,  if  a  sine  wave  of  current, 

/  =  /  cos  B, 


392  ALTERNATING-CURRENT  PHENOMENA. 

passes  through  a  circuit  of  synchronously  varying  reac- 
tance ;  as  for  instance,  the  armature  of  a  unitooth  alterna- 
tor or  synchronous  motor  —  or,  more  general,  an  alternator 
whose  armature  reluctance  is  different  in  different  positions 
with  regard  to  the  field  poles  —  and  the  reactance  is  ex- 
pressed by 


or,  more  general, 

X  = 

the  wave  of  magnetism  is 


X  =  x    1  +  yr  ^  cos  (2  y  ft-  & 

l 


hence  the  wave  of  induced  E.M.F. 


=  *sin/3  +      sin  ()8  -  fflO  + 


[e,  sin  ((2  y  +  1) 
sin  ((2y+  l)/8  -«,+!)]}  ; 

that  is,  the  pulsation  of  reactance  of  frequency,  2y,  intro- 
duces two  higher  harmonics  of  the  order   (2y  —  1),  and 

(2y  +   l\ 

If    ^T=^l 


,  =*{sin0  +  |sinG8-a)  +  .|l  sin  (3/J-o,)^ 

Since  the  pulsation  of  reactance  due  to  magnetic  satu- 
ration and  hysteresis  is  essentially  of  the  frequency,  21V, 


DISTORTION  OF   WAVE-SHAPE.  393 

—  that  is,  describes  a  complete  cycle  for  each  half -wave  of 
current,  —  this  shows  why  the  distortion  of  wave-shape  by 
hysteresis  consists  essentially  of  a  triple  harmonic. 

The  phase  displacement  between  e  and  i,  and  thus  the 
power  consumed  or  produced  in  the  electric  circuit,  depend 
\ipon  the  angle,  o>,  as  discussed  before. 

239.  In  case  of  a  distortion  of  the  wave-shape  by 
reactance,  the  distorted  waves  can  be  replaced  by  their 
equivalent  sine  waves,  and  the  investigation  with  suffi- 
cient exactness  for  most  cases  be  carried  out  under  the 
assumption  of  sine  waves,  as  done  in  the  preceding  chapters. 

Similar  phenomena  take  place  in  circuits  containing 
polarization  cells,  leaky  condensers,  or  other  apparatus 
representing  a  synchronously  varying  negative  reactance. 
Possibly  dielectric  hysteresis  in  condensers  causes  a  dis- 
tortion similar  to  that  due  to  magnetic  hysteresis. 

Pulsation  of  Resistance. 

240.  To  a  certain  extent  the  investigation  of  the  effect 
of  synchronous  pulsation  of  the  resistance  coincides  with 
that  of  reactance ;  since  a  pulsation  of  reactance,  when 
unsymmetrical  with  regard  to  the  current  wave,  introduces 
an  energy  component  which  can  be  represented  by  an 
"  effective  resistance." 

Inversely,  an  unsymmetrical  pulsation  of  the  ohmic 
resistance  introduces  a  wattless  component,  to  be  denoted 
by  "effective  reactance." 

A  typical  case  of  a  synchronously  pulsating  resistance  is 
represented  in  the  alternating  arc. 

The  apparent  resistance  of  an  arc  depends  upon  the 
current  passing  through  the  arc ;  that  is,  the  apparent 

resistance     Of     the     arc     =   Potential  difference^between  electrodes     jg     high 

for  small  currents,  low  for  large  currents.  Thus  in  an 
alternating  arc  the  apparent  resistance  will  vary  during 


304  ALTERNATING-CURRENT  PHENOMENA. 

every  half-wave  of  current  between  a  maximum  value  at 
zero  current  and  a  minimum  value  at  maximum  current, 
thereby  describing  a  complete  cycle  per  half-wave  of  cur- 
rent. 

Let  the  effective  value  of  current  passing  through  the 
arc  be  represented  by  /. 

Then  the  instantaneous  value  of  current,  assuming  the 
current  wave  as  sine  wave,  is  represented  by 

/  =  7V2sin/3; 

and  the  apparent  resistance  of  the  arc,  in  first  approxima- 
tion, by 

R  =  r  (1  +  e  cos  2  j8)  ; 

thus  the  potential  difference  at  the  arc  is 

e  =  iR  =  /V2Vsin/3(l  -f  e  cos  2/3) 


Hence  the  effective  value  of  potential  difference, 


and  the  apparent  resistance  of  the  arc, 


r.-f-ry/t-.  +  f 

The  instantaneous  power  consumed  in  the  arc  is, 


Hence  the  effective  power, 


DISTORTION  OF   WAVE-SHAPE.  395 

The  apparent  power,  or  volt  amperes  consumed  by  the 
arc,  is, 


thus  the  power  factor  of  the  arc, 


that  is,  less  than  unity. 

241.  We  find  here  a  case  of  a  circuit  in  which  the 
power  factor  —  that  is,  the  ratio  of  watts  to  volt  amperes 
—  differs  from  unity  without  any  displacement  of  phase ; 
that  is,  while  current  and  E.M.F.  are  in  phase  with  each 
other,  but  are  distorted,  the  alternating  wave  cannot  be 
replaced  by  an  equivalent  sine  wave ;  since  the  assumption 
of  equivalent  sine  wave  would  introduce  a  phase  displace- 
ment, 

cos  w  =/ 

of  an  angle,  w,  whose  sign  is  indefinite. 

As  an  instance  are  shown,  in  Fig.  173  for  the  constants, 

1=  12 

r=   3 

£    =.9 

the  resistance, 

R  =  3  {I  +  .9  cos  2  /3)  ; 

the  current, 

*    =  17  sin  /3 ; 

tha  potential  difference, 

e  =  28  (sin  ft  +  .82  sin  3  £). 
In  this  case  the  effective  E.M.F.  is 
£=25.5; 


396  ALTERNATING-CURRENT  PHENOMENA. 

the  apparent  resistance, 


the  power, 


the  apparent  power, 


the  power  factor, 


r0  =  2.13  ; 
P  =  244 ; 

El  =307; 
/  =  .796. 


Fig.  173.     Periodically  Varying  Resistance. 

As  seen,  with  a  sine  wave  of  current  the  E.M.F.  wave 
in  an  alternating  arc  will  become  double-peaked,  and  rise 
very  abruptly  near  the  zero  values  of  current.  Inversely, 
with  a  sine  wave  of  E.M.F.  the  current  wave  in  an  alter- 
nating arc  will  become  peaked,  and  very  flat  near  the  zero 
values  of  E.M.F. 

242.  In  reality  the  distortion  is  of  more  complex  nature ; 
since  the  pulsation  of  resistance  in  the  arc  does  not  follow 


DISTORTION  OF   WAVE-SHAPE. 


397 


a  simple  sine  law  of  double  frequency,  but  varies  much 
more  abruptly  near  the  zero  value  of  current,  making 
thereby  the  variation  of  E.M.F.  near  the  zero  value  of 
current  much  more  abruptly,  or,  inversely,  the  variation 
of  current  more  flat. 

A  typical  wave  of  potential  difference,  with  a  sine  wave 
of  current  passing  through  the  arc,  is  given  in  Fig.  174.* 


1     13    13   1     15 


ONE  PAIR  CARBONS 

EG U LATE D  BY  HAND 

A.  C.  dynamo   e.  m.  f 

•'  "        "      current*. 

"   "        "       watts. 


7   18    19    20   S 


Fig.  174.     Electric  Arc. 

243.  The  value  of  e,  the  amplitude  of  the  resistance 
pulsation,  largely  depends  upon  the  nature  of  the  electrodes 
and  the  steadiness  of  the  arc,  and  with  soft  carbons  and  a 
steady  arc  is  small,  and  the  power  factor  f  of  the  arc  near 
unity.  With  hard  carbons  and  an  unsteady  arc,  e  rises 
greatly,  higher  harmonics  appear  in  the  pulsation  of  resis- 
tance, and  the  power  factor  f  falls,  being  in  extreme  cases 
even  as  low  as  .6. 

The  conclusion  to  be  drawn  herefrom  is,  that  photo- 
metric tests  of  alternating  arcs  are  of  little  value,  if,  besides 
current  and  voltage,  the  power  is  not  determined  also  by 
means  of  electro-dynamometers. 

*  From  American  Institute  of  Electrical  Engineers,  Transactions,  1890,  p- 
376.  Tobey  and  Walbridge,  on  the  Stanley  Alternate  Arc  Dynamo. 


398 


A  L  TERN  A  TING-CURRENT  PHENOMENA . 


CHAPTER    XXIII. 

EFFECTS    OF    HIGHER    HARMONICS. 

244.    To  elucidate  the  variation  in  the  shape  of  alternat- 
ing waves  caused  by  various  harmonics,  in  Figs.  175  and 


Fig.  175.    Effect  of  Triple  Harmonic. 


176  are  shown  the  wave-forms  produced  by  the  superposi- 
tion of  the  triple  and  the  quintuple  harmonic  upon  the 
fundamental  sine  wave. 


EFFECTS  OF  HIGHER  HARMONICS.  399 

In  Fig.  175  is  shown  the  fundamental  sine  wave  and 
the  complex  waves  produced  by  the  superposition  of  a  triple 
harmonic  of  30  per  cent  the  amplitude  of  the  fundamental, 
under  the  relative  phase  displacements  of  0°,  45°,  90°,  135°, 
and  180°,  represented  by  the  equations  : 

sin  ft 

sin  ft  —  .3  sin  3  ft 

sin  ft-  .3  sin  (3/3-45°) 

sin  ft  —  .3  sin  (3  ft  —  90°) 

s'm  ft  -  .3  sin  (3  ft  -  135°) 

sin  ft  —  .3  sin  (3/3  —  180°).     • 

As  seen,  the  effect  of  the  triple  harmonic  is  in  the  first 
figure  to  flatten  the  zero  values  and  point  the  maximum 
values  of  the  wave,  giving  what  is  called  a  peaked  wave. 
With  increasing  phase  displacement  of  the  triple  harmonic, 
the  flat  zero  rises  and  gradually  changes  to  a  second  peak, 
giving  ultimately  a  flat-top  or  even  double-peaked  wave  with 
sharp  zero.  The  intermediate  positions  represent  what  is 
called  a  saw-tooth  wave. 

In  Fig.  176  are  shown  the  fundamental  sine  wave  and 
the  complex  waves  produced  by  superposition  of  a  quintuple 
harmonic  of  20  per  cent  the  amplitude  of  the  fundamental, 
under  the  relative  phase  displacement  of  0°,  45°,  90°,  135°, 
180°,  represented  by  the  equations : 

sin  ft 

sin  ft  —  .2  sin  5  ft 
sin/3-  .2  sin  (5,8-45°) 
sin/3-  .2  sin  (5/3-90°) 
smft-  .2  sin  (5/3-  135°) 
sin/3-  .2  sin  (5/8-  180°). 

The  quintuple  harmonic  causes  a  flat -topped  or  even 
double-peaked  wave  with  flat  zero.  With  increasing  phase 
displacement,  the  wave  becomes  of  the  type  called  saw- 
tooth wave  also.  The  flat  zero  rises  and  becomes  a  third 
peak,  while  of  the  two  former  peaks,  one  rises,  the  other 


400 


AL  TERN  A  TING-  CURRENT  PHENOMENA. 


decreases,    and   the   wave    gradually  changes    to   a  triple- 
peaked  wave  with  one  main  peak,  and  a  sharp  zero. 

As  seen,  with  the  triple  harmonic,  flat-top  or  double- 
peak  coincides  with  sharp  zero,  while  the  quintuple  har- 
monic flat-top  or  double-peak  coincides  with  flat  zero. 


Distortion  of  Wave  Shapa 
by  Quintuple  Harmonfc 
Sin./S-.2sin.(5/?-S5j/ 


J 


\J 


Fig.  176.     Effect  of  Quintuple  Harmonic. 


Sharp  peak  coincides  with  flat  zero  in  the  triple,  with 
sharp  zero  in  the  quintuple  harmonic.  With  the  triple  har- 
monic, the  saw-tooth  shape  appearing  in  case  of  a  phase 
difference  between  fundamental  and  harmonic  is  single, 
while  with  the  quintuple  harmonic  it  is  double. 

Thus  in  general,  from  simple  inspection  of  the  wave 
shape,  the  existence  of  these  first  harmonics  can  be  discov- 
ered. Some  characteristic  shapes  are  shown  in  Fig.  177. 


EFFECTS  OF  HIGHER  HARMONICS. 


401 


Sin/?-.225  sinf3/?-180) , 
""-.05  sin/5/3-180) 


Sin./?- 15  sm.(3/?-180). 


Sin./?-.  15' sin  3/?-.1Q  sir 
(5/J-180) 


f/jjr.  777.    So/ne  Characteristic  Wave  Shapes. 

Flat  top  with  flat  zero  : 

sin  /3  —  .15  sin  3  /3  —  .10  sin  5  0. 
Flat  top  with  sharp  zero  : 

sin  0  -  .225  sin  (3  /3  -  180°)  -  .05  sin  (5  /3  -  180°). 
Double  peak,  with  sharp  zero : 

sin  (3  -  .15  sin  (30-  180°)  -  .10  sin  5  /?. 
Sharp  peak  with  sharp  zero  : 

sin  {3  —  .15  sin  3  0  —  .10  sin  (5  (3  —  180°). 

245.  Since  the  distortion  of  the  wave-shape  consists  in 
the  superposition  of  higher  harmonics,  that  is,  waves  of 
higher  frequency,  the  phenomena  taking  place  in  a  circuit 


402  ALTERNATING-CURRENT  PHENOMENA. 

supplied  by  such  a  wave  will  be  the  combined  effect  of  the 
different  waves. 

Thus  in  a  non-inductive  circuit,  the  current  and  the 
potential  difference  across  the  different  parts  of  the  circuit 
are  of  the  same  shape  as  the  impressed  E.M.F.  If  self- 
induction  is  inserted  in  series  to  a  non-inductive  circuit,  the 
self-induction  consumes  more  E.M.F.  of  the  higher  harmon- 
ics, since  the  reactance  is  proportional  to  the  frequency, 
and  thus  the  current  and  the  E.M.F.  in  the  non-inductive 
part  of  the  circuit  shows  the  higher  harmonics  in  a  reduced 
amplitude.  That  is,  self-induction  in  series  to  a  non-induc- 
tive circuit  reduces  the  higher  harmonics  or  smooths  out 
the  wave  to  a  closer  resemblance  with  sine  shape.  In- 
versely, capacity  in  series  to  a  non-inductive  circuit  con- 
sumes less  E.M.F.  at  higher  than  at  lower  frequency,  and 
thus  makes  the  higher  harmonics  of  current  and  of  poten- 
tial difference  in  the  non-inductive  part  of  the  circuit  more 
pronounced  —  intensifies  the  harmonics. 

Self-induction  and  capacity  in  series  may  cause  an  in- 
crease of  voltage  due  to  complete  or  partial  resonance  with 
higher  harmonics,  and  a  discrepancy  between  volt-amperes 
and  watts,  without  corresponding  phase  displacement,  as 
will  be  shown  hereafter. 

246.  In  long-distance  transmission  over  lines  of  notice- 
able inductance  and  capacity,  rise  of  voltage  due  to  reso- 
nance may  occur  with  higher  harmonics,  as  waves  of  higher 
frequency,  while  the  fundamental  wave  is  usually  of  too  low 
a  frequency  to  cause  resonance. 

An  approximate  estimate  of  the  possible  rise  by  reso- 
nance with  various  harmonics  can  be  obtained  by  the  inves- 
tigation of  a  numerical  instance.  Let  in  a  long-distance 
line,  fed  by  step-up  transformers  at  60  cycles, 

The  resistance  drop  in  the  transformers  at  full  load  =  1%. 
The  inductance  voltage  in  the  transformers  at  full  load  =  5% 

with  the  fundamental  wave. 
The  resistance  drop  in  the  line  at  full  load  =  10%. 


EFFECTS  OF  HIGHER   HARMONICS.  403 

The  inductance  voltage  in  the  line  at  full  load  =  20%  with  the 

fundamental  wave. 
The  capacity  or  charging  current  of  the  line  =  20%  of  the  full- 

load  current  /  at  the  frequency  of  the  fundamental. 

The  line  capacity  may  approximately  be  represented  by 
a  condenser  shunted  across  the  middle  of  the  line.  The 
E.M.F.  at  the  generator  terminals  E  is  assumed  as  main- 
tained constant. 

The  E.M.F.  consumed  by  the  resistance  of  the  circuit 
from  generator  terminals  to  condenser  is 

Ir  =  .06  £, 
or,  r  =  .06  -|  . 

The  reactance  E.M.F.  between  generator  terminals  and 
condenser  is,  for  the  fundamental  frequency, 

Ix  =  .15  £, 

-IK  E 

or,  x    =  .15  —  , 

thus  the  reactance  corresponding  to  the  frequency  (2/£  —  1) 
N  of  the  higher  harmonic  is  : 

x(2k-  1)  =.15(2£-  1)  —  . 
The  capacity  current  at  fundamental  frequency  is  : 


hence,  at  the  frequency  :  (2  k  —  1)  N: 

/  =  .2(2£-l)/Z, 
if: 

e'  =  E.M.F.  of  the  (2  k  —  l)th  harmonic  at  the  condenser, 

e  =  E.M.F.  of  the  (2  k  —  l)th  harmonic  at  the  generator  terminals. 

The  E.M.F.  at  the  condenser  is  :  — 

e'  =  V*2  —  iar2  +  ix  (2k  —  V)  • 


404  AL  TERNA  TING-CURRENT  PHENOMENA. 

hence,  substituted  : 


'  l  —  .059856  (2  k  —  I)2  +  .0009  (2  k  —  I)4 

the  rise  of  voltage  by  inductance  and  capacity. 
Substituting  : 

k=     1  2  3  4  56 

or,    2  £  -  1  =     1  3  5  7  9         11 

it  is,  a  =  1.03        1.36        3.76        2.18          .70       .38 

That  is,  the  fundamental  will  be  increased  at  open  circuit 
by  3  per  cent,  the  triple  harmonic  by  36  per  cent,  the 
quintuple  harmonic  by  276  per  cent,  the  septuple  harmonic 
by  118  per  cent,  while  the  still  higher  harmonics  are 
reduced. 

The  maximum  possible  rise  will  take  place  for  : 

=  0,  or,  2,-  1  =  5.77 


That  is,  at  a  frequency  :  N  =  346,  and  a  =  14.4. 

That  is,  complete  resonance  will  appear  at  a  frequency 
between  quintuple  and  septuple  harmonic,  and  would  raise 
the  voltage  at  this  particular  frequency  14.4  fold. 

If  the  voltage  shall  not  exceed  the  impressed  voltage  by 
more  than  100  per  cent,  even  at  coincidence  of  the  maximum 
of  the  harmonic  with  the  maximum  of  the  fundamental, 

the    triple    harmonic   must    be   less   than    70   per   cent  of    the 

fundamental, 
the  quintuple  harmonic  must  be  less  than  26.5  per  cent  of  the 

fundamental, 
the  septuple  harmonic  must  be  less  than  46  per  cent  of  the 

fundamental. 

The  voltage  will  not  exceed  twice  the  normal,  even  at 
a  frequency  of  complete  resonance  with  the  higher  har- 
monic, if  none  of  the  higher  harmonics  amounts  to  more 


EFFECTS  OF  HIGHER  HARMONICS.  405 

than  7  per  cent,  of  the  fundamental.  Herefrom  it  follows 
that  the  danger  of  resonance  in  high  potential  lines  is  in 
general  greatly  over-estimated,  since  the  conditions  assumed 
in  this  instance  are  rather  more  severe  than  found  in  prac- 
tice, the  capacity  current  of  the  line  very  seldom  reaching 
20%  of  the  main  current. 

247.  The  power  developed  by  a  complex  harmonic  wave 
in  a  non-inductive  circuit  is  the  sum  of  the  powers  of  the 
individual  harmonics.      Thus  if  upon  a  sine  wave  of  alter- 
nating E.M.F.  higher  harmonic  waves  are  superposed,  the 
effective  E.M.F.,  and  the  power  produced  by  this  wave  in  a 
given  circuit  or  with  a  given  effective  current,  are  increased. 
In  consequence  hereof  alternators  and  synchronous  motors 
of  ironclad  unitooth  construction  —  that  is,  machines  giving 
waves  with  pronounced  higher  harmonics  —  give  with  the 
same  number  of  turns  on  the  armature,  and  the  same  mag- 
netic flux  per  field  pole  at  the  same  frequency,  a  higher 
output  than  machines  built  to  produce  sine  waves. 

248.  This  explains  an  apparent  paradox  : 

If  in  the  three-phase  star-connected  generator  with  the 
magnetic  field  constructed  as  shown  diagrammatically  in 
Fig.  162,  the  magnetic  flux  per  pole  =  $,  the  number  of 
turns  in  series  per  circuit  =  n,  the  frequency  =  N,  the 
E.M.F.  between  any  two  collector  rings  is: 

E=  V2~7T^2;z<S>10-8. 

since  2«  armature  turns  simultaneously  interlink  with  the 
magnetic  flux  3>. 

The  E.M.F.  per  armature  circuit  is  : 


hence  the  E.M.F.  between  collector  rings,  as  resultant  of 
two  E.M.Fs.  e  displaced  by  60°  from  each  other,  is  : 


406  ALTERNATING-CURRENT  PHENOMENA. 

while  the  same  E.M.F.  was  found  by  direct  calculation 
from  number  of  turns,  magnetic  flux,  and  frequency  to  be 
equal  to  2e;  that  is  the  two  values  found  for  the  same 
E.M.F.  have  the  proportion  V3  :  2  =  1  :  1.154. 


Fig.  178.     Three-phase  Star-connected  Alternator. 

This  discrepancy  is  due  to  the  existence  of  more  pro- 
nounced higher  harmonics  in  the  wave  e  than  in  the  wave 
E  =  e  X  V3,  which  have  been  neglected  in  the  formula  : 


Hence  it  follows  that,  while  the  E.M.F.  between  two  col- 
lector rings  in  the  machine  shown  diagrammatically  in  Fig. 
178  is  only  e  x  V3,  by  massing  the  same  number  of  turns 
in  one  slot  instead  of  in  two  slots,  we  get  the  E.M.F.  2  e 
or  15.4  per  cent  higher  E.M.F.,  that  is,  larger  output. 


EFFECTS   OF  HIGHER   HARMONICS.  407 

It  follows  herefrom  that  the  distorted  E.M.F.  wave  of 
a  unitooth  alternator  is  produced  by  lesser  magnetic  flux  per 
pole  —  that  is,  in  general,  at  a  lesser  hysteretic  loss  in  the 
armature  or  at  higher  efficiency  —  than  the  same  effective 
E.M.F.  would  be  produced  with  the  same  number  of  arma- 
ture turns  if  the  magnetic  disposition  were  such  as  to  pro- 
duce a  sine  wave. 

249.  Inversely,  if  su<:h  a  distorted  wave   of  E.M.F.  is 
impressed  upon  a  magnetic  circuit,  as,  for  instance,  a  trans- 
former, the  wave  of  magnetism  in  the  primary  will  repeat 
in  shape  the  wave  of  magnetism  interlinked  with  the  arma- 
ture coils  of  the  alternator,  and  consequently,  with  a  lesser 
maximum  magnetic  flux,  the  same  effective  counter  E.M.F. 
will  be  produced,  that  is,  the  same  power  converted  in  the 
transformer.     Since  the  hysteretic  loss  in  the  transformer 
depends  upon  the  maximum  value  of  magnetism,  it  follows 
that  the  hysteretic  loss  in  a  transformer  is  less  with  a  dis- 
torted wave  of  a  unitooth  alternator  than  with  a  sine  wave. 

Thus  with  the  distorted  waves  of  unitooth  machines, 
generators,  transformers,  and  synchronous  motors  —  and 
induction  motors  in  so  far  as  they  are  transformers  — 
operate  more  efficiently. 

250.  From    another    side    the    same   problem    can    be 
approached. 

If  upon  a  transformer  a  sine  wave  of  E.M.F.  is  im- 
pressed, the  wave  of  magnetism  will  be  a  sine  wave  also. 
If  now  upon  the  sine  wave  of  E.M.F.  higher  harmonics, 
as  sine  waves  of  triple,  quintuple,  etc.,  frequency  are 
superposed  in  such  a  way  that  the  corresponding  higher 
harmonic  sine  waves  of  magnetism  do  not  increase  the 
maximum  value  of  magnetism,  or  even  lower  it  by  a 
coincidence  of  their  negative  maxima  with  the  positive 
maximum  of  the  fundamental,  —  in  this  case  all  the  power 
represented  by  these  higher  harmonics  of  E.M.F.  will  be 


408  ALTERNATING-CURRENT  PHENOMENA. 

transformed  without  an  increase  of  the  hysteretic  loss,  or 
even  with  a  decreased  hysteretic  loss. 

Obviously,  if  the  maximum  of  the  higher  harmonic  wave 
of  magnetism  coincides  with  the  maximum  of  the  funda- 
mental, and  thereby  makes  the  wave  of  magnetism  more 
pointed,  the  hysteretic  loss  will  be  increased  more  than  in 
proportion  to  the  increased  power  transformed,  i.e.,  the 
efficiency  of  the  transformer  will  be  lowered. 

That  is  :  Some  distorted  waves  of  E.M.F.  are  transformed 
at  a  lesser,  some  at  a  larger,  hysteretic  loss  than  the  sine 
wave,  if  the  same  effective  E.M.F.  is  impressed  upon  the 
transformer. 

The  unitooth  alternator  wave  and  the  first  wave  in  Fig. 
175  belong  to  the  former  class ;  the  waves  derived  from 
continuous-current  machines,  tapped  at  two  equi-distant 
points  of  the  armature,  in  general,  to  the  latter  class. 

251.  Regarding  the  loss  of  energy  by  Foucault  or  eddy 
currents,  this  loss  is  not  affected  by  distortion  of  wave 
shape,  since  the  E.M.F.  of  eddy  currents,  as  induced 
E.M.F.,  is  proportional  to  the  secondary  E.M.F.  ;  and 
thus  at  constant  impressed  primary  E.M.F.,  the  energy 
consumed  by  eddy  currents  bears  a  constant  relation  to 
the  output  of  the  secondary  circuit,  as  obvious,  since  the 
division  of  power  between  the  two  secondary  circuits  — 
the  eddy  current  circuit,  and  the  useful  or  consumer  cir- 
cuit —  is  unaffected  by  wave-shape  or  intensity  of  mag- 
netism. 

252.  In  high  potential  lines,  distorted  waves  whose 
maxima  are  very  high  above  the  effective  values,  as  peaked 
waves,  may  be  objectionable  by  increasing  the  strain  on 
the  insulation.  It  is,  however,  not  settled  yet  beyond 
doubt  whether  the  striking-distance  of  a  rapidly  alternat- 
ing potential  depends  upon  the  maximum  value  or  upon 


EFFECTS  OF  HIGHER  HARMONICS.  409 

some  value  between  effective  and  maximum.  Since  dis- 
ruptive phenomena  do  not  always  take  place  immediately 
after  application  of  the  potential,  but  the  time  element  plays 
ari  important  part,  it  is  possible  that  insulation-strain  and 
striking-distance  is,  in  a  certain  range,  dependent  upon  the 
effective  potential,  and  thus  independent  of  the  wave-shape. 

In  this  respect  it  is  quite  likely  that  different  insulating 
materials  show  a  different  behavior,  and  homogeneous  solid 
substances,  as  paraffin,  depend  in  their  disruptive  strength 
upon  the  maximum  value  of  the  potential  difference,  while 
heterogeneous  materials,  as  mica,  laminated  organic  sub- 
stances, air,  etc.,  that  is  substances  in  which  the  disruptive 
strength  decreases  with  the  time  application  of  the  potential 
difference,  are  less  affected  by  very  high  peaks  of  E.M.F. 
of  very  short  duration. 

In  general,  as  conclusions  may  be  derived  that  the  im- 
portance of  a  proper  wave-shape  is  generally  greatly  over- 
rated, but  that  in  certain  cases  sine  waves  are  desirable, 
in  other  cases  certain  distorted  waves  are  preferable. 


410  ALTERNATING-CURRENT  PHENOMENA. 


CHAPTER   XXIV. 

SYMBOLIC  REPRESENTATION  OF  GENERAL 
ALTERNATING  WAVES. 

253.     The  vector  representation, 

A  =  a1  +y<zu  =  a  (cos  a  -\-j  sin  d) 
of  the  alternating  wave, 

A  —  a0  cos  (<£  —  a) 

applies  to  the  sine  wave  only. 

The  general  alternating  wave,  however,  contains  an  in- 
finite series  of  terms,  of  odd  frequencies, 

A  =  Al  cos  (<£  —  #1)  4-  Az  cos  (3  <£  —  #3)  +  A&  cos  (5  <£  —  #5)  -f 

thus  cannot  be  directly  represented  by  one  complex  vector 
quantity. 

The  replacement  of  the  general  wave  by  its  equivalent 
sine  wave,  as  before  discussed,  that  is  a  sine  wave  of  equal 
effective  intensity  and  equal  power,  while  sufficiently  accu- 
rate in  many  cases,  completely  fails  in  other  cases,  espe- 
cially in  circuits  containing  capacity,  or  in  circuits  containing 
periodically  (and  in  synchronism  with  the  wave)  varying 
resistance  or  reactance  (as  alternating  arcs,  reaction  ma- 
chines, synchronous  induction  motors,  oversaturated  mag- 
netic circuits,  etc.). 

Since,  however,  the  individual  harmonics  of  the  general 
alternating  wave  are  independent  of  each  other,  that  is,  all 
products  of  different  harmonics  vanish,  each  term  can  be 
represented  by  a  complex  symbol,  and  the  equations  of  the 
general  wave  then  are  the  resultants  of  those  of  the  indi- 
vidual harmonics. 


REPRESENTATION  OF  ALTERNATING    WAVES.      411 

This  can  be  represented  symbolically  by  combining  in 
one  formula  symbolic  representations  of  different  frequen- 
cies, thus, 

00 

A  =  £.»-i  (a*  +jn  */) 

i 
where, 

and  the  index  of  the/M  merely  denotes  that  the/s  of  differ- 

entindices  n,  while   algebraically  identical,  physically  rep- 

resent different  frequencies,  and  thus  cannot  be  combined. 

The  general  wave  of  E.M.F.  is  thus  represented  by, 


the  general  wave  of  current  by, 


if, 


is  the  impedance  of  the  fundamental  harmonic,  where 

xm  is  that  part  of  the  reactance  which  is  proportional  to 

the  frequency  (inductance,  etc.). 

x0  is  that  part  of  the  reactance  which  is  independent  of 

the  frequency  (mutual  induction,  synchronous  motion,  etc.). 
xc  is  that  part  of  the  reactance  which  is  inversely  pro- 

portional to  the  frequency  (capacity,  etc.). 

The  impedance  for  the  nth  harmonic  is, 


r  —Jnn  xm 


This  term  can  be  considered  as  the  general  symbolic 
expression  of  the  impedance  of  a  circuit  of  general  wave 
shape. 


412  ALTERNATING-CURRENT  PHENOMENA. 

Ohm's   law,  in  symbolic   expression,  assumes   for  the 
general  alternating  wave  the  form, 

/-Jo, 


E  =  IZ  or, 


Z  =  £or, 


Z  =  r  -n 


The  symbols  of  multiplication  and  division  of  the  terms 
E,  /,  ^f,  thus  represent  not  algebraic  operation,  but  multi- 
plication and  division  of  corresponding  terms  of  E,  T,  Z, 
that  is,  terms  of  the  same  index  «,  or,  in  algebraic  multipli- 
cation and  division  of  the  series  E,  /,  all  compound  terms, 
that  is  terms  containing  two  different  w's,  vanish. 

254.     The  effective  value  of  the  general  wave  : 
a  =  AI  cos  (<£  —  «,)  +  As  cos  (3  <£  —  a8)  +^5  cos  (5  <f>  —  #6)  +.  . 

is  the  square  root  of  the  sum  of  mean  squares  of  individual 
harmonics, 

A=  V  i  {  A?  +  A82  +  A?  +  .  .  .  | 

Since,  as  discussed  above,  the  compound  terms,  of  two 
different  indices  «,  vanish,  the  absolute  value  of  the  general 
alternating  wave, 


REPRESENTATION  OF   ALTERNATING    WAVES.      413 

is  thus, 

A 


which  offers  an  easy  means  of  reduction  from  symbolic  to 
absolute  values. 

Thus,  the  absolute  value  of  the  E.M.F. 


s, 


the  absolute  value  of  the  current, 


is, 


255.  The  double  frequency  power  (torque,  etc.)  equa- 
tion of  the  general  alternating  wave  has  the  same  symbolic 
expression  as  with  the  sine  wave  : 


=  Pl  +JPJ 


1 

where, 


41-4  ALTERNATING-CURRENT  PHENOMENA. 

The  jn  enters  under  the  summation  sign  of  the  "  watt- 
less power  "  1$,  so  that  the  wattless  powers  of  the  different 
harmonics  cannot  be  algebraically  added. 

i        Thus, 

The  total  "  true  power"  of  a  general  alternating  current 
circuit  is  the  algebraic  sum  of  the  powers  of  the  individual 
harmonics. 

The  total  "wattless  power"  of  a  general  alternating 
current  circuit  is  not  the  algebraic,  but  the  absolute  sum  of 
the  wattless  powers  of  the  individual  harmonics. 

Thus,  regarding  the  wattless  power  as  a  whole,  in  the 
general  alternating  circuit  no  distinction  can  be  made  be- 
tween lead  and  lag,  since  some  harmonics  may  be  leading, 
others  lagging. 

The  apparent  power,  or  total  volt-amperes,  of  the  circuit 
is, 


The  power  factor  of  the  circuit  is, 


The  term  "inductance  factor,"  however,  has  no  mean- 
ing any  more,  since  the  wattless  powers  of  the  different 
harmonics  are  not  directly  comparable. 

The  quantity, 


,...._  ...  wattless  power 

has  no  physical  significance,  and  is  not = 

total  apparent  power 


REPRESENTATION  OF  ALTERNATING    WAVES.       4]  > 

The  term,  /#. 

El 

=  2/n~17 


where, 


consists  of  a  series  of  inductance  factors  qn  of  the  individual 
harmonics. 

As  a  rule,  if  <f  =      2^-1  ^n2, 


for  the  general  alternating  wave,  that  is  q  differs  from 

fo=vr^72 

The  complex  quantity, 


Q        El  ~  El 


1 

takes  in  the  circuit  of  the  general  alternating  wave  the 
same  position  as  power  factor  and  inductance  factor  with 
the  sine  wave. 

p 

17=  -~  may  be  called  the  "  circuit  factor  " 

It  consists  of  a  real  term  /,  the  power  factor,  and  a 
series  of  imaginary  terms  jn  qn,  the  inductance  factors  of 
the  individual  harmonics. 


416  ALTERNATING-CURRENT  PHENOMENA. 

The  absolute  value  of  the  circuit  factor  : 


as  a  rule,  is  <  1. 

256.     Some  applications  of  this  symbolism  will  explain 
its  mechanism  and  its  usefulness  more  fully. 

\st  Instance  :     Let  the  E.M.F., 


be  impressed  upon  a  circuit  of  the  impedance, 

7  •      (  *CN 

Z  =  *•—./„  \nxm  -- 


that  is,  containing  resistance  r,  inductive  reactance  xm  and 
capacity  reactance  xc  in  series. 

Let 

e?  =  720  ef  =  540 

V  =  283  4"  =  -  283 

e£  =  -  104  *6"  =  138 

or, 

^  =  900  tan  e^  =        .75 

*,  =  400  tan  o)3  =  -  1 

^5  =  173  tan  w5  =  - 1.33 

It  is  thus  in  symbolic  expression, 

Zj  =  10  +  80/;  *!  =  80.6 

Z3  =  10  zz  =  10 

ZB  =  10  -  32/;  25  =  33.5 

and,  E.M.F., 

^  =  (720  +  540/0  +  (283  -  283y;)  +  (-  104  +  138/5) 

or  absolute, 

E  =  1000 


REPRESENTATION  OF  ALTERNATING    WAVES.      417 


and  current, 

_  £  _  720  +  540/t      283  -  283/8      -  104  +  138./; 
Z~~    10  +  80/i    "  10  10-32y5 


=  (7.76  -  8.04/i)  +  (28.3  -  28.3/8)  +  (-  4.86  -  1.73  A) 

or,  absolute, 

7=41.85 

of  which  is  of  fundamental  frequency,      ll  =  11.15 
"       "      "   "  triple  "  I3  =  40 

«       «   «  quintuple  "  I5  =  5.17 

The  total  apparent  power  of  the  circuit  is, 

Q  =  £7=41,850 
The  true  power  of  the  circuit  is  : 

/»  =  [7i  7]1  =  1240  +  16,000  +  270 

=  17,510 
the  wattless  power, 

j  PJ  =/  [7i  7]J  =  10,000^  -  850/6 
thus,  the  total  power, 

P=  17,510  +  10,000/;  -  850y5 

That  is,  the  wattless  power  of  the  first  harmonic  is 
leading,  that  of  the  third  harmonic  zero,  and  that  of  the  fifth 
harmonic  lagging. 

17,510  =  I2  r,  as  obvious. 
The  circuit  factor  is, 


•        Q        El 

=  .418  +  .239  j\  -  .0203/5 

or,  absolute, 


u  =  V.4182+  .2392  +  .02032 
=  .482 


The  power  factor  is, 

p  =  .418 


418  ALTERNATING-CURRENT  PHENOMENA. 

The  inductance  factor  of  the  first  harmonic  is  :  ql  =  .239, 
that  of  the  third  harmonic  ft  =  0,  and  of  the  fifth  harmonic 
ft  =  -  -0203. 

Considering  the  waves  as  replaced  by  their  equivalent 
sine  waves,  from  the  sine  wave  formula, 

f  +  qf  =  1 
the  inductance  factor  would  be, 

ft  =  -914 
and  the  phase  angle, 

tan  a,  =  ^=  '-^=2.8  «  =  65.4° 

p       .41o 

giving  apparently  a  very  great  phase  displacement,  while  in 
reality,  of  the  41.85  amperes  total  current,  40  amperes  (the 
current  of  the  third  harmonic)  are  in  phase  with  their 
E.M.F. 

We  thus  have  here  a  case  of  a  circuit  with  complex  har- 
monic waves  which  cannot  be  represented  by  their  equiva- 
lent sine  waves.  The  relative  magnitudes  of  the  different 
harmonics  in  the  wave  of  current  and  of  E.M.F.  differ 
essentially,  and  the  circuit  has  simultaneously  a  very  low 
power  factor  and  a  very  low  inductance  factor;  that  is,  a  low 
power  factor  exists  without  corresponding  phase  displace- 
ment, the  circuit  factor  being  less  than  one-half. 

Such  circuits,  for  instance,  are  those  including  alternat- 
ing arcs,  reaction  machines,  synchronous  induction  motors, 
reactances  with  over-saturated  magnetic  circuit,  high  poten- 
tial lines  in  which  the  maximum  difference  of  potential  ex- 
ceeds the  voltage  at  which  brush  discharges  begin,  polariza- 
tion cells,  and  in  general  electrolytic  conductors  above  the 
dissociation  voltage  of  the  electrolyte,  etc.  Such  circuits 
cannot  correctly,  and  in  many  cases  not  even  approxi- 
mately, be  treated  by  the  theory  of  the  equivalent  sine 
waves,  but  require  the  symbolism  of  the  complex  harmonic 
wave. 


REPRESENTATION  OF  ALTERNATING    WAVES.      419 

257.  2d  instance:  A  condenser  of  capacity  C0  =  20 
m.f.  is  connected  into  the  circuit  of  a  60-cycle  alternator 
giving  a  wave  of  the  form, 

e  =  E  (cos  <£  -  .10  cos  3  <£  -  .08  cos  5  <f>  +  .06  cos  7  <£) 
or,  in  symbolic  expression, 

£  =  e(!1-  .10,  -  .085  +  .067) 
The  synchronous  impedance  of  the  alternator  is, 
ZQ  =  r0  —jnnx0  =  .3  —  5  njn 

What  is  the  apparent  capacity  C  of  the  condenser  (as  cal- 
culated from  its  terminal  volts  and  amperes)  when  connected 
directly  with  the  alternator  terminals,  and  when  connected 
thereto  through  various  amounts  of  resistance  and  induc- 
tive reactance. 

The  capacity  reactance  of  the  condenser  is, 
106 


or,  in  symbolic  expression, 

Let 

Z^  =.r  —  jn  nv  =  impedance  inserted  in  series  with  the 
condenser. 

The  total  impedance  of  the  circuit  is  then, 

n 
The  current  in  the  circuit  is, 


(.3  +  r)  -  j  (x  -  132)       (.3  +  r)  -j3  (3  x  -  29) 

^8 ^6 -j 

(.3  +  r)  -j,  (5x-  1.4)       (.3  +  r)  -j\(7x  +  16.1)J 


420  ALTERNATING-CURRENT    PHENOMENA. 

and  the  E.M.F.  at  the  condenser  terminals, 


; 
Jn  V 

4.4  js 


(.3  +  r)  -A  (x  -  132)     (.3  +  r)  -  jz  (3  *  -  29) 

__  2.iiy5  1.13;;  -i 

(.3  +  r)  -j6  (5x-  1.4)  ^  (.3  +  r)  -/7  (7  x  +  16.1)  J 
thus  the  apparent  capacity  reactance  of  the  condenser  is, 


and  the  apparent  capacity, 

106 


^.)     ^r  =  0  :  Resistance  r  in  series  with  the  condenser. 
Reduced  to  absolute  values,  it  is, 

1  .01  .0064  .0036 


17424  19.4 


(.8+r)a+  17424      (.3  +r)2  +  841      (.3  +  r)2  +  1.96      (.3  -f  r)2  +2 

(£.)     r  =  0  :  Inductive  reactance  x  in   series  with  the 
condenser.     Reduced  to  absolute  values,  it  is, 

1  .01  .0064       __  .0036 

—  1.42  "*". 


1.4)2      .09+(7;r-f  16. 


—  132)2      . 

From  —g  are  derived  the  values  of  apparent  capacity, 


c= 


and  plotted  in  Fig.  179  for  values  of  r  and  x  respectively 
varying  from  0  to  22  ohms. 

As  seen,  with  neither  additional  resistance  nor  reactance 
in  series  to  the  condenser,  the  apparent  capacity  with  this 
generator  wave  is  84  m.f.,  or  4.2  times  the  true  capacity, 


REPRESENTATION  OF  ALTERNATING    WAVES.      421 

and  gradually  decreases  with  increasing  series  resistance,  to 
C=  27.5  m.f.  =  1.375  times  the  true  capacity  at  r=  13.2 
ohms,  or  TV  the  true  capacity  reactance,  with  r  =  132  ohms, 
or  with  an  additional  resistance  equal  to  the  capacity  reac- 
tance, C  =  20.5  m.f.  or  only  2.5%  in  excess  of  the  true 
capacity  C0,  and  at  r  =  oo  ,  C =  20,3  m.f.  or  1.5%  in  excess 
of  the  true  capacity. 

With  reactances,  but  no  additional  resistance  r  in  series, 
the  apparent  capacity  C  rises  from  4.2  times  the  true 
capacity  at  x  =  0,  to  a  maximum  of  5,03  times  the  true 
capacity,  or  C=  100.6  m.f.  at  x  =  .28,  the  condition  of  res- 
onance of  the  fifth  harmonic,  then  decreases  to  a  minimum 
of  27  m.f.,  or  35  %  in  excess  of  the  true  capacity,  rises  again 
to  60.2  m.f.,  or  3.01  times  the  true  capacity  at  x  =  9.67, 
the  condition  of  resonance  with  the  third  harmonic,  and 
finally  decreases,  reaching  20  m.f.,  or  the  true  capacity  at 
x  =  132,  or  an  inductive  reactance  equal  to  the  capacity 
reactance,  then  increases  again  to  20.2  m.f.  at  x  =  oo  . 

This  rise  and  fall  of  the  apparent  capacity  is  within  cer- 
tain limits  independent  of  the  magnitude  of  the  higher 
harmonics  of  the  generator  wave  of  E.M.F.,  but  merely  de- 
pends upon  their  presence.  That  is,  with  such  a  reactance 
connected  in  series  as  to  cause  resonance  with  one  of  the 
higher  harmonics,  the  increase  of  apparent  capacity  is  ap- 
proximately the  same,  whatever  the  value  of  the  harmonic, 
whether  it  equals  25%  of  the  fundamental  or  less  than  5%, 
provided  the  resistance  in  the  circuit  is  negligible.  The 
only  effect  of  the  amplitude  of  the  higher  harmonic  is  that 
when  it  is  small,  a  lower  resistance  makes  itself  felt  by  re- 
ducing the  increase  of  apparent  capacity  below  the  value  it 
would  have  were  the  amplitude  greater. 

It  thus  follows  that  the  true  capacity  of  a  condenser 
cannot  even  approximately  be  determined  by  measuring 
volts  and  amperes  if  there  are  any  higher  harmonics  present 
in  the  generator  wave,  except  by  inserting  a  very  large  re- 
sistance or  reactance  in  series  to  the  condenser. 


422 


ALTERNATING-CURRENT  PHENOMENA. 


258.     §d  instance :    An  alternating  current  generator 

of  the  wave, 

E.  =  2000  [lt  +  .12,  -  .23B  -  .13,] 

and  of  synchronous  impedance, 

Z0  =  .3-5*/; 
feeds  over  a  line  of  impedance, 


C4PJ 

CITV 

Co  = 

=  20 

mf  i 

CM 

CL'IT 

OF 

r,E\ 

HAT 

R 

1 

8 

=  EI  O-J--I.L—  .ya-t-uc/  OF 

Zo^S-S),   n    WITH    RESIS 

fASC 

DANCE 

k  r(I) 

! 

c 

R    RE 

ACT 

NCE 

*^ 

I)     1 

SE 

!ES 

C: 

£ 

100 

/\ 

0 

90 

J 

i 

^ft 

I 

k 

5 

rn 

I 

\ 

\ 

i 

H 

^ 

/ 

\ 

.w 

\ 

\ 

/ 

X 

10 

REE 

X 

STAC 

ii 

^=^~ 

CE  r 

= 

;=" 

^ 

^ 

= 

REA( 

—     - 
TAN!1 

X 

•-  

^S 

•^ 

*  , 

;  

— 

= 

^= 

_» 

=3<F 

10 

I  ; 

, 

i      !'o     ! 

1 

2     1 

1 

1 

1 

1 

-     1 

-t       1 

r,        2, 

1 

0 

a  synchronous  motor  of  the  wave, 

EI  =  2250  [(cos  oj  +/i  sin  «)  +  .24  (cos  3  w  -(-y's  sin  3  o>)] 
and  of  synchronous  impedance, 

Z2  =  .3  -  C  «/; 

The  total  impedance  of  the  system  is  then, 
Z  =  ZQ  +  Zl  +  Z2 
=  2.6-15«/n 


REPRESENTATION  OF  ALTERNATING    WAVES.      423 

thus  the  current, 


_  2000  -  2250  cos  o>  -  2250/\  sin  o>      240  -  540  cos  3a>  -  540/;  sin  3a> 
2.6  -  15/i  2.6  -  45y8 

460  260 

~~  2.6  -  75  j\      2.6  -  105  jj 

=  « 

where, 

aj1  =  22.5  -  25.2  cos  co  +  146  sin  a> 

ag1  =  .306  -  .69  cos  3  to  +  11.9  sin  3 

a,1  =  -  .213 

«7i  =  -  .061 

V1  =  130  -  146  cos  w  -  25.2  sin  a> 

^8«  =  5.3  -  11.9  cos  3  o>  -  .69  sin  3  o> 

a*  =  -  6.12 

a7u  =  -  2.48 

or,  absolute, 

1st  harmonic, 

3d  harmonic, 

5th  harmonic, 

a6  =  6.12 
7th  harmonic, 

«7  =  2.48 


/=   V 
while  the  total  current  of  higher  harmonics  is, 


424  ALTERNATING-CURRENT  PHENOMENA. 

The  true  input  of  the  synchronous  motor  is, 


=  (  2250  a£  cos  o>  +  2250  a?  sin  o>  )  +  (  540  a?  cos  3o>  +  540  asn  sin  3o>) 

=  /V  +  /'s1 
^  =  2250  (a?  cos  <o  +  af  sin  o>) 


.  780.    Synchronous  Motor, 


REPRESENTATION  OF  ALTERNATING    WAVES.      425 

is  the  power  of  the  fundamental  wave, 

P£  =  540  (a,,1  cos  3  w  +  as11  sin  3  o>) 

the  power  of  the  third  harmonic. 

The  5th  and  7th  harmonics  do  not  give  any  power, 
since  they  are  not  contained  in  the  synchronous  motor 
wave.  Substituting  now  different  numerical  values  for  u> 
the  phase  angle  between  generator  E.M.F.  and  synchronous 
motor  counter  E.M.F.,  corresponding  values  of  the  currents 
/ 70,  and  the  powers  P\  P*,  /Y  are  derived.  These  are 
plotted  in  Fig.  180  with  the  total  current  /as  abcissae.  To 
each  value  of  the  total  current  /  correspond  two  values  of 
the  total  power  P\  a  positive  value  plotted  as  Curve  I. — 
synchronous  motor  —  and  a  negative  value  plotted  as 
Curve  II. — alternating  current  generator — .  Curve  III. 
gives  the  total  current  of  higher  frequency  I0,  Curve  IV., 
the  difference  between  the  total  current  and  the  current  of 
fundamental  frequency,  / — alt  in  percentage  of  the  total 
current  /,  and  V  the  power  of  the  third  harmonic,  Pj,  in 
percentage  of  the  total  power  P1. 

Curves  III.,  IV.  and  V.  correspond  to  the  positive  or 
synchronous  motor  part  of  the  power  curve  P\  As  seen, 
the  increase  of  current  due  to  the  higher  harmonics  is 
small,  and  entirely  disappears  at  about  180  amperes.  The 
power  of  the  third  harmonic  is  positive,  that  is,  adds  to  the 
work  of  the  synchronous  motor  up  to  about  140  amperes, 
or  near  the  maximum  output  of  the  motor,  and  then  becomes 
negative. 

It  follows  herefrom  that  higher  harmonics  in  the  E.M.F. 
waves  of  generators  and  synchronous  motors  do  not  repre- 
sent a  mere  waste  of  current,  but  may  contribute  more  or 
less  to  the  output  of  the  motor.  Thus  at  75  amperes  total 
current,  the  percentage  of  increase  of  power  due  to  the 
higher  harmonic  is  equal  to  the  increase  of  current,  or  in 
other  words  the  higher  harmonics  of  current  do  work  with 
the  same  efficiency  as  the  fundamental  wave. 


426  ALTERNATING-CURRENT  PHENOMENA. 

259.  kth  Instance:  In  a  small  three-phase  induction 
motor,  the  constants  per  delta  circuit  are 

Primary  admittance  Y=  .002  +  .03/ 

Self-inductive  impedance  ZQ  =  Zl  =  .6  —  2.4/ 

and  a  sine  wave  of  E.M.F.  e0  =  110  volts  is  impressed  upon 
the  motor. 

The  power  output  P,  current  input  7S,  and  power  factor 
/,  as  function  of  the  slip  s  are  given  in  the  first  columns  of 
the  following  table,  calculated  in  the  manner  as  described  in 
the  chapter  on  Induction  Motors. 

To  improve  the  power  factor  of  the  motor  and  bring  it 
to  unity  at  an  output  of  500  watts,  a  condenser  capacity  is 
required  giving  4.28  amperes  leading  current  at  110  volts, 
that  is,  neglecting  the  energy  loss  in  the  condenser,  capacity 
susceptance 


In  this  case,  let  Is  =  current  input  into  the  motor  per 
delta  circuit  at  slip  s,  as  given  in  the  following  table. 

The  total  current  supplied  by  the  circuit  with  a  sine 
wave  of  impressed  E.M.F.,  is 

/i  =  ls  -  4.28/ 

energy  current 
and  heref  rom  the  power  factor  =  -  ;  —         —  ,  given  in 

total  current 
the  second  columns  of  the  table. 

If  the  impressed  E.M.F.  is  not  a  sine  wave  but  a  wave 
of  the  shape 

E,  =  e,  (lx  +  .12.  -  .235  -  .134,) 

to  give  the  same  output,  the  fundamental  wave  must  be  the 
same  :  e0  =  110  volts,  when  assuming  the  higher  harmonics 
in  the  motor  as  wattless,  that  is 

£0  =  110,  +  13.2,  -  25.3B  -  14.7, 

=  *o  +  £<? 
where  £0l  =  13.2,  -  25.3B  -  14.7T 

=  component  of  impressed  E.M.F.  of  higher  frequency- 


REPRESENTATION^  Of  ALTERNATING    WAVES.      427 

The  effective  value  is  : 

EQ  =  114.5  volts. 

The  condenser  admittance  for  the  general  alternating 
wave  is 

Yc=  -.039«/; 

Since  the  frequency  of  rotation  of  the  motor  is  very 
small  compared  with  the  frequency  of  the  higher  harmonics, 
as  total  impedance  of  the  motor  for  these  higher  harmonics 
can  be  assumed  the  stationary  impedance,  and  by  neglecting 
the  resistance  it  is 

Z1  =  -  njn  (XQ  +  XJ 

=  -  4.8  njn 

The  exciting  admittance  of  the  motor,  for  these  higher 
harmonics,  is,  by  neglecting  the  conductance, 


n 
and  the  higher  harmonics  of  counter  E.M.F. 


Thus  we  have, 


Current  input  in  the  condenser, 

fc  =  E,  Yc 

=  -  4.28/i  -  1.54/3  +  4.93/5  +  4.02/7 

High  frequency  component  of  motor  impedance  current, 

|£  =  .92/3  -  1.06y5  -  .44/7 
High  frequency  component  of  motor  exciting  current, 


=  .07/3  -  -08/5  -  . 


428 


AL  TERN  A  TING-CURRENT  PHEA'OAIENA. 


thus,  total  high  frequency  component  of  motor  current, 

/o1  =  |f  +  &  y1 

=  .99y3  -  1.14,;  -  .47/7 
and  total  current, 

without  condenser, 

4  =  4  +  41 

=  Is  +  .99/3  -  1.14,;  -  .47/7 

with  condenser, 


=  4  -  4.28,i  -  . 
and  herefrom  the  power  factor. 


3.79,;  +  3.55/7 


T    PER  PHASE 


In  the  following  table  and  in  Fig.  181  are  given  the 
values  of  current  and  power  factor  :  — 

I.  With  sine  wave  of  E.M.F.,  of  110  volts,  and  no  condenser. 

II.  With  sine  wave  of  E.M.F ,  of  1 10  volts,  and  with  condenser. 

III.  With  distorted  wave  of  E.M.F.,  of  114.6  volts,  and  no  condenser. 

IV.  With  distorted  wave  of  E.M.F.,  of  114.5  volts,  and  with  condenser. 


REPRESENTATION  OF  ALTERNATING    WAVES.       429 


f 
0 
.01 
.02 
.035 
.05 
.07 
.10 
.13 
.15 

P 

0 
160 
320 
500 
660 
810 
885 
900 
890 

I, 
.24+   3.10/ 
1.73+   3.16/ 
3.32+   3.47> 
5.16+   4.28/ 
6.95+    5.4/ 
8.77+   7.3; 
10.1    +   9.85/ 
10.45  +  11.45/ 
10.75  +  12.9/ 

It 
3.1 
3.6 
4.8 
6.7 
8.8 
11.4 
14.1 
15.5 
16.8 

7.8 
48 
69 
77 
79 
77 
71.5 
67.5 
64 

f  

1.2 
2.1 
3.4 
5.2 
7.0 
9.3 
11.5 
12.7 
13.8 

—  •> 
P 
20 
84 
97.2 
100 
98.7 
94.5 
87 
82 
78 

i  — 

3.5 
3.9 
5.1 
6.9 
8.9 
11.5 
14.2 
15.6 
16.9 

1  —  \ 

6.6 
43 
64 
72.5 
76 
73.5 
68 
64.5 
61 

/  
I 

5.2 
5.5 
6.1 
7.2 
8.6 
10.6 
12.6 
13.7 
14.7 

—  i 

4. 
81 
64 

(18 
7T 
80 
7T 
73: 
7Q/ 

The  curves  II.  and  IV.  with  condenser  are  plotted  in 
dotted  lines  in  Fig.  181.  As  seen,  even  with  such  a  dis- 
torted wave  the  current  input  and  power  factor  of  the  motor 
are  not  much  changed  if  no  condenser  is  used.  When  using 
a  condenser  in  shunt  to  the  motor,  however,  with  such  a 
wave  of  impressed  E.M.F.  the  increase  of  the  total  current, 
due  to  higher  frequency  currents  in  the  condenser,  is  greater 
than  the  decrease,  due  to  the  compensation  of  lagging  cur- 
rents, and  the  power  factor  is  actually  lowered  by  the  con- 
denser, over  the  total  range  of  load  up  to  overloads,  and 
especially  at  light  loads. 

Where  a  compensator  or  transformer  is  used  for  feeding- 
the  condenser,  due  to  the  internal  self-induction  of  the  com- 
pensator, the  higher  harmonics  of  current  are  still  more 
accentuated,  that  is  the  power  factor  still  more  lowered. 

In  the  preceding  the  energy  loss  in  the  condenser  and 
compensator  and  that  due  to  the  higher  harmonics  of  cur- 
rent in  the  motor  has  been  neglected.  The  effect  of  this 
energy  loss  is  a  slight  decrease  of  efficiency  and  correspond- 
ing increase  of  power  factor.  The  power  produced  by  the 
higher  harmonics  has  also  been  neglected ;  it  may  be  posi- 
tive or  negative,  according  to  the  index  of  the  harmonic, 
and  the  winding  of  the  motor  primary.  Thus  for  instance, 
the  effect  of  the  triple  harmonic  is  negative  in  the  quarter- 
phase  motor,  zero  in  the  three-phase  motor,  etc.,  altogether,, 
however,  the  effect  of  these  harmonics  is  very  small. 


430  ALTERNATING-CURRENT  PHENOMENA. 


CHAPTER    XXV. 

GENERAL   POLYPHASE    SYSTEMS. 

260.  A  polyphase  system  is  an  alternating-current  sys- 
tem in  which  several  E.M.Fs.  of  the  same  frequency,  but 
displaced  in  phase  from  each  other,  produce  several  currents 
of  equal  frequency,  but  displaced  phases. 

Thus  any  polyphase  system  can  be  considered  as  con- 
sisting of  a  number  of  single  circuits,  or  branches  of  the 
polyphase  system,  which  may  be  more  or  less  interlinked 
with  each  other. 

In  general  the  investigation  of  a  polyphase  system  is 
carried  out  by  treating  the  single-phase  branch  circuits 
independently. 

Thus  all  the  discussions  on  generators,  synchronous 
motors,  induction  motors,  etc.,  in  the  preceding  chapters, 
apply  to  single-phase  systems  as  well  as  polyphase  systems, 
in  the  latter  case  the  total  power  being  the  sum  of  the 
powers  of  the  individual  or  branch  circuits. 

If  the  polyphase  system  consists  of  n  equal  E.M.Fs. 
displaced  from  each  other  by  1  /  n  of  a  period,  the  system 
is  called  a  symmetrical  system,  otherwise  an  unsymmetrical 
system. 

Thus  the  three-phase  system,  consisting  of  three  equal 
E.M.Fs.  displaced  by  one-third  of  a  period,  is  a  symmetrical 
system.  The  quarter-phase  system,  consisting  of  two  equal 
E.M.Fs.  displaced  by  90°,  or  one-quarter  of  a  period,  is  an 
unsymmetrical  system. 

261.  The  flow  of   power  in  a  single-phase  system  is 
pulsating ;  that  is,  the  watt  curve  of  the  circuit  is  a  sine 


GENERAL  POLYPHASE  SYSTEMS,  431 

wave  of  double  frequency,  alternating  between  a  maximum 
value  and  zero,  or  a  negative  maximum  value.  In  a  poly- 
phase system  the  watt  curves  of  the  different  branches  of 
the  system  are  pulsating  also.  Their  sum,  however,  or  the 
total  flow  of  power  of  the  system,  may  be  either  constant 
or  pulsating.  In  the  first  case,  the  system  is  called  a 
balanced  system,  in  the  latter  case  an  unbalanced  system. 

The  three-phase  system  and  the  quarter-phase  system, 
with  equal  load  on  the  different  branches,  are  balanced  sys- 
tems ;  with  unequal  distribution  of  load  between  the  indi- 
vidual branches  both  systems  become  unbalanced  systems. 


Fig.   181. 


Fig.   182. 

The  different  branches  of  a  polyphase  system  may  be 
either  independent  from  each  other,  that  is,  without  any 
electrical  interconnection,  or  they  may  be  interlinked  with 
each  other.  In  the  first  case,  the  polyphase  system  is 
called  an  independent  system,  in  the  latter  case  an  inter- 
linked system. 

The  three-phase  system  with  star-connected  or  ring-con- 
nected generator,  as  shown  diagrammatically  in  Figs.  181 
and  182,  is  an  interlinked  system. 


432 


ALTERNATING-CURRENT  PHENOMENA. 


The  four-phase  system  as  derived  by  connecting  four 
equidistant  points  of  a  continuous-current  armature  with 
four  collector  rings,  as  shown  diagrammatically  in  Fig.  183, 


Fig.   183. 

is  an  interlinked  system  also.  The  four-wire  quarter-phase 
system  produced  by  a  generator  with  two  independent 
armature  coils,  or  by  two  single-phase  generators  rigidly 
connected  with  each  other  in  quadrature,  is  an  independent 
system.  As  interlinked  system,  it  is  shown  in  Fig.  184,  as 
star-connected  four-phase  system. 


-E 


r 


Fig.  184. 

262.    Thus,  polyphase  systems  can  be  subdivided  into : 
Symmetrical  systems  and  unsymmetrical  systems. 
Balanced  systems  and  unbalanced  systems. 
Interlinked  systems  and  independent  systems. 
The  only  polyphase  systems  which  have  found  practical 
application  are : 

The  three-phase  system,  consisting  of  three  E.M.Fs.  dis- 


GENERAL   POLYPHASE  SYSTEMS.  433 

placed  by  one-third  of  a  period,  used  exclusively  as  inter- 
linked system. 

The  quarter-phase  system,  consisting  of  two  E.M.Fs.  in 
quadrature,  and  used  with  four  wires,  or  with  three  wires, 
which  may  be  either  an  interlinked  system  or  an  indepen- 
dent system. 

The  six-phase  system,  consisting  of  two  three-phase  sys- 
tems in  opposition  to  each  other,  and  derived  by  transforma- 
tion from  a  three-phase  system,  in  the  alternating  supply 
circuit  of  large  synchronous  converters. 

The  inverted  three-phase  system,  consisting  of  two 
E.M.F.'s  displaced  from  each  other  by  60°,  and  derived 
from  two  phases  of  a  three-phase  system  by  transformation 
with  two  transformers,  of  which  the  secondary  of  one  is 
reversed  with  regard  to  its  primary  (thus  changing  the 
phase  difference  from  120°  to  180°  -  120°  =  60°),  finds  a 
limited  application  in  low  tension  distribution. 


434  ALTERNATING-CURRENT  PHENOMENA. 


CHAPTER   XXVI. 

SYMMETRICAL  POLYPHASE    SYSTEMS. 

263.  If  all  the  E.M.Fs.  of  a  polyphase  system  are  equal 
in  intensity,  and  differ  from  each  other  by  the  same  angle 
of  difference  of  phase,  the  system  is  called  a  symmetrical 
polyphase  system. 

Hence,  a  symmetrical  w-phase  system  is  a  system  of  n 
E.M.Fs.  of  equal  intensity,  differing  from  each  other  in 
phase  by  1  /  n  of  a  period  : 

*i  =  E  sin  (3 ; 
e2=£sm((3-^L\', 


en  =  E  sin  (  ft  -  L  V*  ~    - 
\ 

The  next  E.M.F.  is  again  : 

^  =  E  sin  (ft  —  2  TT)  =  E  sin  ft. 

In  the  polar  diagram  the  n  E.M.Fs.  of  the  symmetrical 
0-phase  system  are  represented  by  n  equal  vectors,  follow- 
ing each  other  under  equal  angles. 

Since  in  symbolic  writing,  rotation  by  l/«  of  a  period, 
or  angle  2ir/n,  is  represented  by  multiplication  with : 


the  E.M.Fs.  of  the  symmetrical  polyphase  system  are: 


SYMMETRICAL  POLYPHASE  SYSTEMS.  435 


/  9  T-  ?  -rr 

E(  cos  —  +  /  sin  —    = 
•  ' 


n 

„  f        2  (n  —  1)  TT   .     .  .    2  («  —  1) 
^  f  cos  —  -i  -  L  --  \-j  sm  —  ^  -  ^ 

'    V  » 

The  next  E.M.F.  is  again  : 

E  (  cos  2  -n-  +j  sin  2  TT)  =  .£  e"  =  .£. 
Hence,  it  is 

27T       .        •     .       27T  n/? 

e  =  cos  -  -  -f  J  sm  -  =  V  1. 
;z  « 

Or  in  other  words  : 

In  a  symmetrical  «-phase  system  any   E.M.F.   of   the 
system  is  expressed  by  : 

e'-Ej 

where  :  e  =  -y/1. 

264.    Substituting  now  for  n  different  values,  we  get 
the  different  symmetrical  polyphase  systems,  represented  by 

*E\ 

,  n/T  2  7T  .     .       2  7T 

where,  e  =  vl  =  cos  --  \-j  sin  —  •  . 

n  n 

1.)    «  =  1     e  =  1     c«'^  =  .£, 
the  ordinary  single-phase  system. 

2.)    «  =  2    e  =  -  1    J  £  =  £  and  -  £. 

Since   —  ^  is  the  return  of  E,  n  =  2  gives  again  the 
single-phase  system. 


3 
-1-/V3 


436  ALTERNATING-CURRENT  PHENOMENA. 

The  three  E.M.Fs.  of  the  three-phase  system  are  : 


-i-yV3 


Consequently  the  three-phase  system  is  the  lowest  sym- 
metrical polyphase  system. 

4.)    n  =  4,   c  =  cos  —  +/  sin  —  =/,   £2  =  —  1,   e3  =  -  /. 
4  4 

The  four  E.M.Fs.  of  the  four-phase  system  are: 

*£  =  £,    J£,     -E,     -JE. 
They  are  in  pairs  opposite  to  each  other  : 
E  and  —  E  •  j  E  and  —JE. 

Hence  can  be  produced  by  two  coils  in  quadrature  with 
each  other,  analogous  as  the  two-phase  system,  or  ordinary 
alternating-current  system,  can  be  produced  by  one  coil. 

Thus  the  symmetrical  quarter-phase  system  is  a  four- 
phase  system. 

Higher  systems,  than  the  quarter-phase  or  four-phase 
system,  have  not  been  very  extensively  used,  and  are  thus 
of  less  practical  interest.  A  symmetrical  six-phase  system, 
derived  by  transformation  from  a  three-phase  system,  has 
found  application  in  synchronous  converters,  as  offering  a 
higher  output  from  these  machines,  and  a  symmetrical  eight- 
phase  system  proposed  for  the  same  purpose. 

265.  A  characteristic  feature  of  the  symmetrical  »- 
phase  system  is  that  under  certain  conditions  it  can  pro- 
duce a  M.M.F.  of  constant  intensity. 

If  «  equal  magnetizing  coils  act  upon  a  point  under 
equal  angular  displacements  in  space,  and  are  excited  by  the 
n  E.M.Fs.  of  a  symmetrical  w-phase  system,  a  M.M.F.  of 
constant  intensity  is  produced  at  this  point,  whose  direction 
revolves  synchronously  with  uniform  velocity. 

Let, 
n'  =•  number  of  turns  of  each  magnetizing  coil. 


SYMMETRICAL  POLYPHASE  SYSTEMS.  437 

E=  effective  value  of  impressed  E.M.F. 
/  =  effective  value  of  current. 

Hence, 
&  =n'f=  effective  M.M.F.  of  one  of  the  magnetizing  coils. 

Then  the  instantaneous  value  of  the  M.M.F.  of  the  coil 
acting  in  the  direction  2  «•*'/»  is  : 


The  two  rectangular  space  components  of  this  M.M.F.  are  ; 


and 


Hence  the  M.M.F.  of  this  coil  can  be  expressed  by  the 
symbolic  formula  : 


fi 

n       \          n 

Thus  the  total  or  resultant  M.M.F.  of  the  n  coils  dis- 
placed under  the  n  equal  angles  is  : 


or,  expanded  : 


n 


438  ALTERNATING-CURRENT  PHENOMENA. 

It  is,  however : 


cos'2  —  +  / sin  —  cos  —  =  £  (  1  +  cos  —  +/ sin  —] 

n  n  n  V  w  w    / 

\  / 

sin 2=1  cos  ?Z£+ysin«2=£=  ^Yl  -  cos  i^'-ysin4^' 

«          »  •  «       z  y  «  « 

_  ^   /I     _   ,2A  X 


2(1-^ 

and,  since: 

5t<2<  =  0, 

it  is,  /=  nn'f^  (-sin  ft  _  y  cos  ft), 

or, 


the  symbolic  expression  of  the  M.M.F.  produced  by  the 
«  circuits  of  the  symmetrical  «-phase  system,  when  exciting 
n  equal  magnetizing  coils  displaced  in  space  under  equal 
angles. 

The  absolute  value  of  this  M.M.F.  is  : 

nn'  I       n"S        n  <5 


V2        V2  2 

Hence  constant  and  equal  w/V2  times  the  effective 
M.M.F.  of  each  coil  or  «/2  times  the  maximum  M.M.F. 
of  each  coil. 

The  phase  of  the  resultant  M.M.F.  at  the  time  repre- 
sented by  the  angle  ft  is  : 

tan  w  =  —  cot  /8  ;  hence  w  =  /?  —  ^ 

That  is,  the  M.M.F.  produced  by  a  symmetrical  «-phase 
system  revolves  with  constant  intensity  : 


SYMMETRICAL  POLYPHASE  SYSTEMS.  439 

F=  —  • 

V25 

and  constant  speed,  in  synchronism  with  the  frequency  of 
the  system  ;  and,  if  the  reluctance  of  the  magnetic  circuit 
is  constant,  the  magnetism  revolves  with  constant  intensity 
and  constant  speed  also,  at  the  point  acted  upon  symmetri- 
cally by  the  n  M.M.Fs.  of  the  w-phase  system. 

This  is  a  characteristic  feature  of  the  symmetrical  poly- 
phase system. 

266.  In  the  three-phase  system,  n  =  3,  F=  1.5  <5max 
where  $max  is  the  maximum  M.M.F.  of  each  of  the  magne- 
tizing coils. 

In  a  symmetrical  quarter-phase  system,  n  =  4,  F  =  2 
^tnax,  where  $maje  is  the  maximum  M.M.F.  of  each  of  the 
four  magnetizing  coils,  or,  if  only  two  coils  are  used,  since 
the  four-phase  M.M.Fs.  are  opposite  in  phase  by  two,  F  = 
&max>  where  ^max  is  the  maximum  M.M.F.  of  each  of  the 
two  magnetizing  coils  of  the  quarter-phase  system. 

While  the  quarter-phase  system,  consisting  of  two  E.M.Fs. 
displaced  by  one-quarter  of  a  period,  is  by  its  nature  an 
unsymmetrical  system,  it  shares  a  number  of  features  — 
as,  for  instance,  the  ability  of  producing  a  constant  result- 
ant M.M.F.  —  with  the  symmetrical  system,  and  may  be 
considered  as  one-half  of  a  symmetrical  four-phase  system. 

Such  systems,  consisting  of  one-half  of  a  symmetrical 
system,  are  called  hemisymmetrical  systems. 


440  ALTERNATING-CURRENT  PHENOMENA. 


CHAPTER    XXVII. 

BALANCED    AND    UNBALANCED    POLYPHASE    SYSTEMS. 

267.    If  an  alternating  E.M.F.  : 

e  =  E  V2  sin  (3, 
produces  a  current  : 

*  =  7V2sin  (/?  —  a), 

where  u>  is  the  angle  of  lag,  the  power  is  : 

p  =  ei  =  2  £Ssin  ft  sin  (ft  —  S) 

=  £S(cos  a  —  cos  (2  £  —  a)), 

and  the  average  value  of  power  : 


Substituting  this,  the  instantaneous  value  of   power  is 
found  as  : 


Hence  the  power,  or  the  flow  of  energy,  in  an  ordinary 
single-phase  alternating-current  circuit  is  fluctuating,  and 
varies  with  twice  the  frequency  of  E.M.F.  and  current, 
unlike  the  power  of  a  continuous-current  circuit,  which  is 

constant  : 

/-** 

If  the  angle  of  lag  £  =  0  it  is  : 

p  =  P  (1  —  cos  2  0)  ; 

hence  the  flow  of  power  varies  between  zero  and  2  Pt  where 
P  is  the  average  flow  of  energy  or  the  effective  power  of 
the  circuit. 


BALANCED  POLYPHASE  SYSTEMS.  441 

If  the  current  lags  or  leads  the  E.M.F.  by  angle  £  the 
power  varies  between 

and 


cos  u> 

that  is,  becomes  negative  for  a  certain  part  of  each  half- 
wave.  That  is,  for  a  time  during  each  half-wave,  energy 
flows  back  into  the  generator,  while  during  the  other  part 
of  the  half-wave  the  generator  sends  out  energy,  and  the 
difference  between  both  is  the  effective  power  of  the  circuit. 
If  £  =  90°,  it  is : 

O    rt   , 

"  p  > 


that  is,  the  effective  power  :  P  =  0,  and  the  energy  flows 
to  and  fro  between  generator  and  receiving  circuit. 

Under  any  circumstances,  however,  the  flow  of  energy  in 
the  single-phase  system  is  fluctuating  at  least  between  zero 
and  a  maximum  value,  frequently  even  reversing. 

268.    If  in  a  polyphase  system 

*D  ez>  *s>  •   •  •  •  =  instantaneous  values  of  E.M.F. ; 
h)  *2,  t'a,  •  •  •  •  =  instantaneous  values  of  current  pro- 
duced thereby ; 

the  total  flow  of  power  in  the  system  is  : 

p  =  glt\  -f  <?2/2  -j-  e,j,  -f  .  .  .  . 
The  average  flow  of  power  is  : 

P  =  £i  /i  cos  £>!  -(-  E<i  /2  cos  w2  -f-  .  .  .  . 

The  polyphase  system  is  called  a  balanced  system,  if  the 
flow  of  energy  : 

/  =  e\i\  +  <V2  +  W,  +.'.'.;.. 

is  constant,  and  it  is  called  an  unbalanced  system  if  the 
flow  of  energy  varies  periodically,  as  in  the  single-phase  sys- 
tem ;  and  the  ratio  of  the  minimum  value  to  the  maximum 
value  of  power  is  called  the  balance  factor  of  the  system. 


442  ALTERNATING-CURRENT  PHENOMENA. 

Hence  in  a  single-phase  system  on  non-inductive  circuit, 
that  is,  at  no-phase  displacement,  the  balance  factor  is  zero ; 
and  it  is  negative  in  a  single-phase  system  with  lagging  or 
leading  current,  and  becomes  =  —  1,  if  the  phase  displace- 
ment is  90°  — that  is,  the  circuit  is  wattless. 

269.  Obviously,  in  a  polyphase  system  the  balance  of 
the  system  is  a  function  of  the  distribution  of  load  between 
the  different  branch  circuits. 

A  balanced  system  in  particular  is  called  a  polyphase 
system,  whose  flow  of  Energy  is  constant,  if  all  the  circuits 
are  loaded  equally  with  a  load  of  the  same  character,  that 
is,  the  same  phase  displacement. 

270.  All  the  symmetrical  systems  from  the  three-phase 
system  upward  are  balanced  systems.     Many  unsymmetrical 
systems  are  balanced  systems  also. 

1.)    Three-phase  system  : 
Let 

^  =  E  V2  sin  ft,  and     t\  =  I V2  sin  (ft  —  w)  ; 

ez  =  E  V2  sin  (ft  -  120),  /2  =  /  V2  sin  (0  -  «  -  120)  ; 

ez  =  E  V2  sin  (ft  -  240),  /3  =  /  V2  sin  (ft  -  &  -  240) ; 

be  the  E.M.Fs.  of  a  three-phase  system,  and  the  currents 
produced  thereby. 

Then  the  total  flow  of  power  is  : 

/  =  2  .57  (sin  {3  sin  (ft  —  fi)  +  sin  ((3  —  120)  sin  (ft  —  &  —  120) 

+  sin  (ft  —  240)  sin  ($  —  <*  —  240)) 
=  3  .£7 cos  w  =  T5,  or  constant. 

Hence  the  symmetrical  three-phase  system  is  a  balanced 
system. 

2.)    Quarter-phase  system  : 

Let     £l  =  £^2s\nft,  t\  =  I  \/2  sin  (ft  -  5)  ; 

e2  =  E  V2  cos  ft,  4  =  7  V2  cos  (ft  -  £)  ; 


BALANCED  POLYPHASE  SYSTEMS.  443 

be  the  E.M.Fs.  of  the  quarter-phase  system,  and  the  cur- 
rents produced  thereby. 

This  is  an  unsymmetrical"  system,  but  the  instantaneous 
flow  of  power  is  : 

/  =  2  £I(sm  J3  sin  (/?  —  5)  +  cos  ft  cos  (0  —  £>)) 
=  2  £Scos  w  =  P,  or  constant. 

Hence  the  quarter-phase  system  is  an  unsymmetrical  bal- 
anced system. 

3.)  The  symmetrical  «-phase  system,  with  equal  load 
and  equal  phase  displacement  in  all  n  branches,  is  a  bal- 
anced system.  For,  let : 

e(  =  E  V2  sin  ( ft  -  — "\  =  E.M.F. ; 
V  »    / 

/  2    IT    A 

*',-  =  7V2  sin  O  —  S —    =  current 

V  » V 

the  instantaneous  flow  of  power  is : 


l        V  «  7       \  » 

EI  \  yr  cos  a  -57-035^2 /?-£-  — 


or  p  =  n  E I  cos  w  =  T7,  or  constant. 

271.  An  unbalanced  polyphase  system  is  the  so-called 
inverted  three-phase  system,*  derived  from  two  branches  of 
a  three-phase  system  by  transformation  by  means  of  two 
transformers,  whose  secondaries  are  connected  in  opposite 
direction  with  respect  to  their  primaries.  Such  a  system 
takes  an  intermediate  position  between  the  Edison  three- 
wire  system  and  the  three-phase  system.  It  shares  with 
the  latter  the  polyphase  feature,  and  with  the  Edison  three- 

*  Also  called  "polyphase  monocyclic  system,"  since  the  E.M.F.  triangle  is  similar  to 
that  usual  in  the  single-phase  monocyclic  system. 


444  ALTERNATING-CURRENT  PHENOMENA. 

wire  system  the  feature  that  the  potential  difference  be- 
tween the  outside  wires  is  higher  than  between  middle 
wire  and  outside  wire. 

By  such  a  pair  of  transformers  the  two  primary  E.M.Fs. 
of  120°  displacement  of  phase  are  transformed  into  two 
secondary  E.M.Fs.  differing  from  each  other  by  60°.  Thus 
in  the  secondary  circuit  the  difference  of  potential  between 
the  outside  wires  is  V3  times  the  difference  of  potential 
between  middle  wire  and  outside  wire.  At  equal  load  on 
the  two  branches,  the  three  currents  are  equal,  and  differ 
from  each  other  by  120°,  that  is,  have  the  same  relative 
proportion  as  in  a  three-phase  system.  If  the  load  on 
one  branch  is  maintained  constant,  while  the  load  of  the 
other  branch  is  reduced  from  equality  with  that  in  the 
first  branch  down  to  zero,  the  current  in  the  middle  wire 
first  decreases,  reaches  a  minimum  value  of  87  per  cent  of 
its  original  value,  and  then  increases  again,  reaching  at  no 
load  the  same  value  as  at  full  load. 

The  balance  factor  of  the  inverted  three-phase  system 
on  non-inductive  load  is  .333. 

272.  In  Figs.  185  to  192  are  shown  the  E.M.Fs.  as 
e  and  currents  as  i  in  drawn  lines,  and  the  power  as  /  in 
dotted  lines,  for : 


Fig.   185.    Single-phase  System  on  Non-inductive  Load. 

Balance  Factor,  0. 


BALANCED  POLYPHASE  SYSTEMS.  445 


Fig.   186.    Single-phase  System  on  Inductiue  Load  of  60°  Lag. 

Balance  Factor,  -  .333. 


Fig.   187.    Quarter-phase  System  on  Non-inductiui  Load. 

Balance  Factor,  + 1. 


Fig.  183.    Quarter-phase  System  on  Inductiue  Lozd  of  60°  Lag. 
Balance  Factor,  + 1. 


446  ALTERNATING-CURRENT  PHENOMENA. 


Fig.   189.     Three-phase  System  on  Non-induct'we  Load. 

Balance  Factor,  + 1. 


Fig.  190.     Three-phase  System  on  Inductive  Load  of  60°  Lag. 

Balance  Factor,  + 1. 


Fig.   191.     Inverted  Three-phase  System 
on  Non-inductive  Load. 


Balance  Factor, +  .333 


BALANCED  POLYPHASE  SYSTEMS. 


447 


Fig.  174.     Inverted  Three-phase  System  on 

Inductive  Load  of  60°  Lag. 

Balance  Factor,  0. 

273.  The  flow  of  power  in  an  alternating-current  system 
is  a  most  important  and  characteristic  feature  of  the  system, 
and  by  its  nature  the  systems  may  be  classified  into  : 

Monocyclic  systems,  or  systems  with  a  balance  factor  zero 
or  negative. 

Polycyclic  systems,  with  a  positive  balance  factor. 

Balance  factor  —  1  corresponds  to  a  wattless  circuit, 
balance  factor  zero  to  a  non-inductive  single-phase  circuit, 
balance  factor  +  1  to  a  balanced  polyphase  system. 

274.  In   polar   coordinates,   the    flow   of   power  of    an 
alternating-current  system  is  represented  by  using  the  in- 
stantaneous flow  of  power  as  radius  vector,  with  the  angle 
($  corresponding  to   the  time  as   amplitude,   one  complete 
period  being  represented  by  one  revolution. 

In  this  way  the  power  of  an  alternating-current  system 
is  represented  by  a  closed  symmetrical  curve,  having  the 
zero  point  as  quadruple  point.  In  the  monocyclic  systems 
the  zero  point  is  quadruple  nodal  point ;  in  the  polycyclic 
system  quadruple  isolated  point. 

Thus  these  curves  are  sextics.  « 


448  ALTERNATING-CURRENT  PHENOMENA. 

Since  the  flow  of  power  in  any  single-phase  branch  of 
the  alternating-current  system  can  be  represented  by  a  sine 
wave  of  double  frequency  : 


the  total  flow  of  power  of  the  system  as  derived  by  the 
addition  of  the  powers  of  the  branch  circuits  can  be  rep- 
resented in  the  form  : 

/  =  />(!  +  «  sin  (2  £-  a.)) 

This  is  a  wave  of  double  frequency  also,  with  c  as  ampli- 
tude of  fluctuation  of  power. 

This  is  the  equation  of  the  power  characteristics  of  the 
system  in  polar  coordinates. 

275.  To  derive  the  equation  in  rectangular  coordinates 
we  introduce  a  substitution  which  revolves  the  system  of 
coordinates  by  an  angle  o>o/2,  so  as  to  make  the  symmetry 
axes  of  the  power  characteristic  the  coordinate  axes. 


hence,  sin  (2  ft  -  S>0)  =  2  sin  ^  -  ^  )  cos  (/?  -  ^  j  = 
substituted, 

^M'  +  ^j. 

or,  expanded  : 

—  P2  (x2  +  /*  +  2  e  A:^)2  =  0, 


the  sextic  equation  of  the  power  characteristic. 
Introducing : 

a  =  (!  +  «)/'=  maximum  value  of  power, 
b  =  (1  —  c)  P'=  minimum  value  of  power; 


BALANCED  POLYPHASE  SYSTEMS.  449 

it  is  **?> 


a  +  b 
hence,  substituted,  and  expanded  : 

(*»+/)»  -  \{a  (x  +  j)2  +  b  (x  -X>T>  =  0 

the  equation   of  the  power  characteristic,   with  the  main 
power  axes  a  and  b,  and  the  balance  factor:  b  I  a. 
It  is  thus  : 

Single-phase    non-inductive    circuit  :  /  =  />  (1  +  sin  2  <£), 
b  =  0,     a  =  2P 


Single-phase  circuit,  60°  lag  :  /  =  P  (1  +  2  sin  2  <£), 

i*.~+" 


Single-phase  circuit,  90°  lag  :/  =  ^  /sin  2  <£,     b  =  —  E  I, 

a  =  +  El 


2/,  &/a=  -1. 
Three-phase  non-inductive  circuit  :  p  =  P,     ^  =  1,    a  = 

x^+y*  —  P2  =  0:  circle.     &  /  a  =  +  1. 
Three-phase  circuit,  60°  lag  :  /  =  P,     6  =  1,     a  =  1 

a?  +/-  7>a  =  0  :  circle.     £/«=  +  !. 
Quarter-phase  non-inductive  circuit  :p  =  P,b  =  ]-)    a  = 

x*  _|_  y»  _  ^2  =  o  .  circlei     ^  /  ^  =  _|_  i. 

Quarter-phase  circuit,  60°  lag  :  p  =  P,     b  =  1,     tf  =  1 


450  ALTERNATING-CURRENT  PHENOMENA. 

Inverted  three-phase  non-inductive  circuit : 


Inverted  three-phase  circuit  60°  lag  :/  =  f  (1  -\-  sin  2  <£), 
b  =  0,  a  =  2  P 

(y?  +  /)3   _   />2  (• x  _|_  yy   =   0<         fila   =   Qf 

a  and  <5  are  called  the  main  power  axes  of  the  alternating- 
current  system,  and  the  ratio  b  [a  is  the  balance  factor  of 
the  system. 


Figs.   193  and  104.    Power  Characteristic  of  Single-phase  System,  at  60°  and  0°  Lag. 

276.  As  seen,  the  flow  of  power  of  an  alternating-cur- 
rent system  is  completely  characterized  by  its  two  main 
power  axes  a  and  b. 

The  power  characteristics  in   polar  coordinates,    corre- 


BALANCED   POLYPHASE  SYSTEM. 


451 


spending  to  the  Figs.  185,  186,  191,  and  192  are  shown  in 
Figs.  193,  194,  195,  and  196. 


Figs.   195  and  196.    Power  Characteristic  of  Inverted  Three-phase  System,   at  0°  and 
60°  Lag. 

The  balanced  quarter-phase  and  three-phase  systems  give 
as  polar  characteristics  concentric  circles. 


452  ALTERNATING-CURRENT  PHENOMENA. 


CHAPTER    XXVIII. 

INTERLINKED   POLYPHASE    SYSTEMS. 

277.  In  a  polyphase   system   the  different  circuits  of 
displaced  phases,  which  constitute  the  system,  may  either 
be  entirely  separate  and  without  electrical  connection  with 
each   other,   or   they  may   be   connected   with    each   other 
electrically,  so  that  a  part  of  the  electrical  conductors  are 
in  common  to   the  different  phases,  and  in  this  case  the 
system  is  called  an  interlinked  polyphase  system. 

Thus,  for  instance,  the  quarter-phase  system  will  be 
called  an  independent  system  if  the  two  E.M.Fs.  in  quadra- 
ture with  each  other  are  produced  by  two  entirely  separate 
coils  of  the  same,  or  different  but  rigidly  connected,  arma- 
tures, and  are  connected  to  four  wires  which  energize  inde- 
pendent circuits  in  motors  or  other  receiving  devices.  If 
the  quarter-phase  system  is  derived  by  connecting  four 
equidistant  points  of  a  closed-circuit  drum  or  ring-wound 
armature  to  the  four  collector  rings,  the  system  is  an  inter- 
linked quarter-phase  system. 

Similarly  in  a  three-phase  system.  Since  each  of  the 
three  currents  which  differ  from  each  other  by  one-third 
of  a  period  is  equal  to  the  resultant  of  the  other  two  cur- 
rents, it  can  be  considered  as  the  return  circuit  of  the  other 
two  currents,  and  an  interlinked  three-phase  system  thus 
consists  of  three  wires  conveying  currents  differing  by  one- 
third  of  a  period  from  each  other,  so  that  each  of  the  three 
currents  is  a  common  return  of  the  other  two,  and  inversely. 

278.  In  an  interlinked  polyphase  system  two  ways  exist 
of  connecting  apparatus  into  the  system. 


INTERLINKED  POLYPHASE  SYSTEMS. 


453 


1st.  The  star  connection,  represented  diagrammatically 
in  Fig.  197.  In  this  connection  the  n  circuits  excited  by 
currents  differing  from  each  other  by  1  /  n  of  a  period,  are 
connected  with  their  one  end  together  into  a  neutral  point 
or  common  connection,  which  may  either  be  grounded  or 
connected  with  other  corresponding  neutral  points,  or  insu- 
lated. 

In  a  three-phase  system  this  connection  is  usually  called 
a  Y  connection,  from  a  similarity  of  its  diagrammatical  rep- 
resentation with  the  letter  Y,  as  shown  in  Fig.  181. 


2d.  The  ring  connection,  represented  diagrammatically 
in  Fig.  198,  where  the  n  circuits  of  the  apparatus  are  con- 
nected with  each  other  in  closed  circuit,  and  the  corners 
or  points  of  connection  of  adjacent  circuits  connected  to 
the  n  lines  of  the  polyphase  system.  In  a  three-phase 
system  this  connection  is  called  the  delta  connection,  from 
the  similarity  of  its  diagrammatic  representation  with  the 
Greek  letter  Delta,  as  shown  in  Fig.  182. 

In  consequence  hereof  we  distinguish  between  star- 
connected  and  ring-connected  generators,  motors,  etc.,  or 


454  ALTERNATING-CURRENT  PHENOMENA. 


Fig.  198. 


in    three-phase  systems   Y- connected    and    delta-connected 
apparatus. 

279.  Obviously,  the  polyphase  system  as  a  whole  does 
not  differ,  whether  star  connection  or  ring  connection  is 
used  in  the  generators  or  other  apparatus ;  and  the  trans- 
mission line  of  a  symmetrical  «-phase  system  always  con- 
sists of  n  wires  carrying  current  of  equal  strength,  when 
balanced,  differing  from  each  other  in  phase  by  l/«  of  a 
period.  Since  the  line  wires  radiate  from  the  n  terminals 
of  the  generator,  the  lines  can  be  considered  as  being  in 
star  connection. 

The  circuits  of  all  the  apparatus,  generators,  motors, 
etc.,  can  either  be  connected  in  star  connection,  that  is, 
between  one  line  and  a  neutral  point,  or  in  ring  connection, 
that  is,  between  two  adjacent  lines. 

In  general  some  of  the  apparatus  will  be  arranged  in 
star  connection,  some  in  ring  connection,  as  the  occasion 
may  require. 


INTERLINKED  POLYPHASE  SYSTEMS.  455 

280.  In  the  same  way  as  we  speak  of  star  connection 
and  ring  connection  of  the  circuits  of  the  apparatus,  the 
term  star  potential  and  ring  potential,  star  current  and  ring 
current,  etc.,  are  used,  whereby  as  star   potential  or  in  a 
three-phase  circuit  Y  potential,  the  potential  difference  be- 
tween one  of  the  lines  and  the  neutral  point,  that  is,  a  point 
having  the  same  difference  of  potential  against  all  the  lines, 
is  understood  ;  that  is,  the  potential  as  measured  by  a  volt- 
meter  connected  into  star  or  Y  connection.     By   ring   or 
delta  potential    is   understood    the  difference   of    potential 
between  adjacent   lines,  as  measured  by  a  voltmeter  con- 
nected between   adjacent   lines,    in  -ring   or  delta   connec- 
tion. 

In  the  same  way  the  star  or  Y  current  is  the  current 
flowing  from  one  line  to  a  neutral  point  ;  the  ring  or  delta 
current,  the  current  flowing  from  one  line  to  the  other. 

The  current  in  the  transmission  line  is  always  the  star 
or  Y  current,  and  the  potential  difference  between  the  line 
wires,  the  ring  or  delta  potential. 

Since  the  star  potential  and  the  ring  potential  differ 
from  each  other,  apparatus  requiring  different  voltages  can 
be  connected  into  the  same  polyphase  mains,  by  using  either 
star  or  ring  connection. 

281.  If  in  a  generator  with  star-connected  circuits,  the 
E.M.F.    per  circuit  =  E,  and   the   common  connection  or 
neutral  point  is  denoted  by  zero,  the  potentials  of  the  n 
terminals  are  : 


or  in  general  :  t*  JS, 

at  the  z'th  terminal,  where  : 

*  =  0,  1,  2  ....»-  1,     e  =  cos  —  +j  sin  —  =  -\/l. 


456  ALTERNATING-CURRENT  PHENOMENA. 

Hence  the  E.M.F.  in  the  circuit  from  the  zth  to  the  £* 
terminal  is  : 

Eki  =  **  E  —  ^E  =  (c*  —  e')  E. 

The  E.M.F.  between  adjacent  terminals  i  and  i  +  1  is  : 

(e.+i  -J)E  =  e*  (e  -  1)  E. 

In  a  generator  with  ring-connected  circuits,  the  E.M.F. 

per  circuit  : 

cl  E 

is  the  ring  E.M.F.,  and  takes  the  place  of 


while  the  E.M.F.  between  terminal   and  neutral  point,  or 
the  star  E.M.F.,  is  : 


Hence  in  a  star-connected  generator  with  the  E.M.F. 
E  per  circuit,  it  is  : 

Star    E.M.F.,  IE. 

RingE-M.F.,  c'Xc-1)^. 

E.M.F.  between  terminal  /  and  terminal  k,  (c*  —  e')  E. 

In  a  ring-connected  generator  with  the  E.M.F.  E  per 
circuit,  it  is  : 

Star    E.M.F.,  —  ^—  E. 
e  —  1  ' 

Ring  E.M.F.,  C  E. 

E.M.F.  between  terminals  *  and  k,  e   ~  e*  E. 

£  —   1      ' 

In  a  star-connected  apparatus,  the  E.M.F.  and  the  cur- 
rent per  circuit  have  to  be  the  star  E.M.F.  and  the  star 
current.  In  a  ring-connected  apparatus  the  E.M.F.  and 
current  per  circuit  have  to  be  the  ring  E.M.F.  and  ring 
current. 

In  the  generator  of  a  symmetrical  polyphase  system,  if  : 
c''  E  are  the  E.M.Fs.  between  the  n  terminals  and  the 
neutral  point,  or  star  E.M.Fs., 


INTERLINKED  POLYPHASE  SYSTEMS.  457 

If  =  the  currents  issuing  from  terminal  i  over  a  line  of 
the  impedance  Z{  (including  generator  impedance  in  star 
connection),  we  have  : 

Potential  at  end  of  line  i  : 


Difference  of  potential  between  terminals  k  and  i  : 


where  /,.  is  the  star  current  of  the  system,  Zt  the  star  im- 
pedance. 

The  ring  potential  at  the  end  of  the  line  between  ter- 
minals i  and  k  is  Eik,  and  it  is  : 

Eile  =  —  Eti. 

If  now  Iik  denotes  the  current  passing  from  terminal  i  to 
terminal  k,  and  Zik  impedance  of  the  circuit  between  ter- 
minal i  and  terminal  k,  where  : 

fit  =  ~  /*,, 
Zt*  =  Zti, 

it  is  Eik  =  ZitIik. 

If  Iio  denotes  the  current  passing  from  terminal  i  to  a 
ground  or  neutral  point,  and  Zio  is  the  impedance  of  this 
circuit  between  terminal  i  and  neutral  point,  it  is  : 

Eio  =  €*£-  ZiSi  =  Ziolio. 

282.    We  have  thus,  by  Ohm's  law  and  Kirchhoff  's  law  : 

If  *'  E  is  the  E.M.F.  per  circuit  of  the  generator,  be- 
tween the  terminal  i  and  the  neutral  point  of  the  generator, 
or  the  star  E.M.F. 

/,-  =  the  current  issuing  from  the  terminal  i  of  the  gen- 
erator, or  the  star  current. 

Zt  =  the  impedance  of  the  line  connected  to  a  terminal 
i  of  the  generator,  including  generator  impedance. 

EL  =  the  E.M.F.  at  the  end  of  line  connected  to  a  ter- 
minal i  of  the  generator. 


458  ALTERNATING-CURRENT  PHENOMENA. 

Eik  =  the  difference  of  potential  between  the  ends  of 
the  lines  i  and  k. 

Iik  =  the  current  passing  from  line  i  to  line  k. 

Zik  =  the  impedance  of  the  circuit  between  lines  i  and  k. 

Iio,  Iioo  .  .  .  .  =  the  current  passing  from  line  i  to  neu- 
tral points  0,  00,  .... 

Zio,  Zioo  .  .  .  .  =  the  impedance  of  the  circuits  between 
line  i  and  neutral  points  0,  00,  .... 

It  is  then  : 


Zio  =  Zoi,  etc. 
2.)    Et    =JE-ZiIi. 

3.)    Ei    =  Zi0fi0  =  Zioofj00  =  .  .  .  . 

4.)    Eik  =  Et'-  E{  =  (t*  -  e')  E  -  (Zklk  -  ZJ^). 

5.)    Eik  =  ZikIik. 


7.)  If  the  neutral  point  of  the  generator  does  not  exist, 
as  in  ring  connection,  or  is  insulated  from  the  other  neutral 
points  : 


IE/,,  =0; 

n 

5E/ioo  =  0,  etc. 
1 

Where  0,  00,  etc.,  are  the  different  neutral  points  which 
are  insulated  from  each  other. 

If  the  neutral  point  of  the  generator  and  all  the  other 
neutral  points  are  grounded  or  connected  with  each  other, 
it  is: 


INTERLINKED   POLYPHASE  SYSTEMS.  459 

If  the  neutral  point  of  the  generator  and  all  other  neu- 
tral points  are  grounded,  the  system  is  called  a  grounded 
system.  If  the  neutral  points  are  not  grounded,  the  sys- 
tem is  an  insulated  polyphase  system,  and  an  insulated 
polyphase  system  with  equalizing  return,  if  all  the  neutral 
points  are  connected  with  each  other. 

8.)    The  power  of  the  polyphase  system  is  — 


P  =  ^f   e1'  E  Ii  cos  $i  at  the  generator 

1 

•f  =  "^i    ^*  Eik  Iik  cos  <f>it  in  the  receiving  circuits. 


4GO  ALTERNATING-CURRENT  PHENOMENA. 


CHAPTER    XXIX. 

TRANSFORMATION    OF    POLYPHASE    SYSTEMS. 

283.  In  transforming  a  polyphase  system  into  another 
polyphase  system,   it  is   obvious  that  the  primary  system 
must  have  the  same  flow  of  power  as  the  secondary  system, 
neglecting  losses  in  transformation,  and  that  consequently 
a  balanced  system  will  be  transformed  again  in  a  balanced 
system,  and  an  unbalanced  system  into  an  unbalanced  sys- 
tem of  the  same  balance  factor,  since  the  transformer  is  an 
apparatus  not  able  to  store  energy,  and  thereby  to  change 
the   nature  of  the  flow  of  power.      The  energy  stored  as 
magnetism,  amounts  in  a  well-designed  transformer  only  to 
a  very  small  percentage  of  the  total  energy.      This  shows 
the  futility  of    producing  symmetrical  balanced   polyphase 
systems  by  transformation  from  the  unbalanced  single-phase 
system  without   additional  apparatus  able  to  store  energy 
efficiently,  as  revolving  machinery. 

Since  any  E.M.F.  can  be  resolved  into,  or  produced  by, 
two  components  of  given  directions,  the  E.M.Fs.  of  any 
polyphase  system  can  be  resolved  into  components  or  pro- 
duced from  components  of  two  given  directions.  This  en- 
ables the  transformation  of  any  polyphase  system  into  any 
other  polyphase  system  of  the  same  balance  factor  by  two 
transformers  only. 

284.  Let   Elt    E2,    Ez  .  .  .  .  be   the  E.M.Fs.   of  the 
primary  system  which  shall  be  transformed  into  — 

E{,  £2',  £s'  .  .  .  .  the  E.M.Fs.  of  the  secondary 
system. 

Choosing  two  magnetic   fluxes,   <£  and   <£,    of  different 


TRANSFORMATION  OF  POLYPHASE  SYSTEMS,     461 

phases,  as  magnetic  circuits  of  the  two  transformers,  which 
induce  the  E.M.Fs.,  e  and  ?,  per  turn,  by  the  law  of  paral- 
lelogram the  E.M.Fs.,  Elf  E^,  .  .  .  .  can  be  dissolved  into 
two  components,  El  and  Elt  E^  and  Ez,  ....  of  the  phases* 
"e  and  J. 
Then,  - 

E!,  £2,  •  •  '  •  are  the  counter  E.M.Fs.  which  have  to  be- 
induced  in  the  primary  circuits  of  the  first  transformer;. 

Ev  E2,  ....  the  counter  E.M.F.'s  which  have  to  be  in- 
duced in  the  primary  circuits  of  the  second  transformer.. 

hence 

EI  1 7,  £2 1 J  .  .  .  .  are  the  numbers  of  turns  of  the  primary 
coils  of  the  first  transformer. 

Analogously 

EI  /T     £2  IT  .  .  .  .  are  the  number  of  turns  of  the  primary  coils 
in  the  second  transformer. 

In  the  same  manner  as  the  E.M.Fs.  of  the  primary 
system  have  been  resolved  into  components  in  phase  with 
J  and  FJ  the  E.M.Fs.  of  the  secondary  system,  E-^>  E^,  .... 

are  produced  from  components,  E-f  and  E^,  E£  and  EJ, 
....  in  phase  with  ~e  and  J,  and  give  as  numbers  of  second 
ary  turns, — 

£il  /  J,  £2l  /?»••••  in  the  first  transformer ; 
EI  1 7,  EZ  /  F,  ....  in  the  second  transformer. 

That  means  each  of  the  two  transformers  m  and  m  con- 
tains in  general  primary  turns  of  each  of  the  primary 
phases,  and  secondary  turns  of  each  of  the  secondary 
phases.  Loading  now  the  secondary  polyphase  system  in 
any  desired  manner,  corresponding  to  the  secondary  cur- 
rents, primary  currents  will  flow  in  such  a  manner  that  the 
total  flow  of  power  in  the  primary  polyphase  system  is  the 


4j^  ALTERNATING-CURRENT  PHENOMENA. 

same  as  the  total  flow  of  power  in  the  secondary  system, 
plus  the  loss  of  power  in  the  transformers. 

285.  As  an  instance  may  be  considered  the  transforma- 
tion of  the  symmetrical  balanced  three-phase  system 

E  sin  ft,     E  sin  (ft  —  120),      E  sin  (ft  —  240), 
in  an  unsymmetrical  balanced  quarter-phase  system : 

E'  sin  ft,     E'  sin  (ft  —  90). 
Let  the  magnetic  flux  of  the  two  transformers  be 

(/>  cos  £   and    </>  cos  (ft  —  90). 

Then  the  E.M.Fs.  induced  per  turn  in  the  transformers 
e  sin  ft   and   e  sin  (ft  —  90)  ; 

hence,  in  the  primary  circuit  the  first  phase,  E  sin  ft,  will 
give,  in  the  first  transformer,  E/e  primary  turns;  in  the 
second  transformer,  0  primary  turns. 

The  second  phase,  E  sin  (ft  —  120),  will  give,  in  the 
first  transformer,  —  E  /  2  e  primary  turns;  in  the  second 

E  x  ~\/3 

transformer,  — — primary  turns. 

2  e 

The  third  phase,  E  sin  (ft  —  240),  will  give,  in  the  first 
transformer,  —  E /le  primary  turns;  in  the  second  trans- 
former, — primary  turns. 

2  e 

In  the  secondary  circuit  the  first  phase  E'  sin  ft  will  give 
in  the  first  transformer:  E' / e  secondary  turns;  in  the 
second  transformer  :  0  secondary  turns. 

The  second  phase  :  E'  sin  (ft  —  90)  will  give  in  the  first 
transformer  :  0  secondary  turns ;  in  the  second  transformer, 
E' I  e  secondary  turns. 

Or,  if : 

E  =  5,000     E'  =  100,     e  =  10. 


TRANSFORMATION  OF  POLYPHASE   SYSTEMS.       463 


PRIMARY. 
1st.  2d. 


SECONDARY. 
3d.          1st.       2d.        Phase. 


first  transformer 
second  transformer 


+  500 
0 


-  250     -  250 

4-  433     -  433 


10 
0 


0 
10    turns. 


That  means  : 

Any  balanced  polyphase  system  *.jm  be  transformed  by  two 
transformers  only,  without  storage  of  energy,  into  any  other 
balanced  polyphase  system. 

286.  Some  of  the  more  common  methods  of  transfor- 
mation between  polyphase  systems  are  : 


Fig.   799. 

1.  The  delta -Y  connection  of  transformers  between 
three-phase  systems,  shown  in  Fig.  199.  One  side  of  the 
transformers  is  connected  in  delta,  the  other  in  Y.  This 
arrangement  becomes  necessary  for  feeding  four  wires 


rwi  nnr 
V 


Fig.  200. 


three-phase  secondary  distributions.  The  Y  connection  of 
the  secondary  allows  to  bring  out  a  neutral  wire,  while  the 
delta  connection  of  the  primary  maintains  the  balance  be- 
tween the  phases  at  unequal  distribution  of  load. 


464 


ALTERNA TING-CURRENT  PHENOMENA. 


2.  The  L  connection  of  transformers  between  three-phase 
systems,  consisting  in  using  two  sides  of  the  triangle  only, 
as  shown  in  Fig.  200.  This  arrangement  has  the  disadvan- 
tage of  transforming  one  phase  by  two  transformers  in 
series,  hence  is  less  efficient,  and  is  liable  to  unbalance  the 
system  by  the  internal  impedance  of  the  transformers. 


Fig.  201. 

3.    The   main   and  teaser,    or    T  connection  of   trans- 
formers between  three-phase  systems,  as  shown  in  Fig.  201. 

V3 
One  of  the  two  transformers  is  wound  for  ~-~-  times  the 

voltage  of  the  other  (the  altitude  of  the  equilateral  triangle), 
and  connected  with  one  of  its  ends  to  the  center  of  the 


Fig.  202. 

other  transformer.  From  the  point  £  inside  of  the  teaser 
transformer,  a  neutral  wire  can  be  brought  out  in  this  con- 
nection. 

4.  The  monocyclic  connection,  transforming  between 
three-phase  and  inverted  three-phase  or  polyphase  mono- 
cycle,  by  two  transformers,  the  secondary  of  one  being 
reversed  regarding  its  primary,  as  shown  in  Fig.  202. 


TRANSFORMATION  OF  POLYPHASE  SYSTEMS.       465 


5.  The  L  connection  for  transformation  between  quar- 
ter-phase and  three-phase  as  described  in  the  instance,  para- 
graph 257. 

6.  The  T  connection  of  transformation  between  quarter- 
phase  and  three-phase,  as  shown  in  Fig.  203.     The  quar- 
ter-phase side  of  the  transformers  contains  two  equal  and 


Fig.  203. 

independent  (or  interlinked)  coils,  the  three-phase  side  two 

Vs 

coils  with  the  ratio  of  turns  1  -=-  — ^   connected  in  T. 

7.  The  double  delta  connection  of  transformation  from 
three-phase  to  six-phase,  shown  in  Fig.  204.  Three  trans- 
formers, with  two  secondary  coils  each,  are  used,  one  set  of 


Fig    204. 


secondary  coils  connected  in  delta,  the  other  set  in  delta 
also,  but  with  reversed  terminals,  so  as  to  give  a  reversed 
E.M.F.  triangle.  These  E.M.F.'s  thus  give  topographically 
a  six-cornered  star. 


466 


AL  TERN  A  TING-CURRENT  PHENOMENA. 


8.  The   double    Y  connection  of   transformation   from 
three-phase  to  six-phase,  shown  in  Fig.  205.     It  is  analo- 
gous to  (7),  the  delta  connection  merely  being  replaced  by 
the  Y  connection.     The  neutrals  of   the  two    F's  may  be 
connected  together  and  to  an  external  neutral  if  desired. 

9.  The    double   T  connection   of   transformation    from 


Fig.  205. 

three-phase  to  six-phase,  shown  in  Fig.  206.  Two  trans- 
formers are  used  with  two  secondary  coils  which  are  T  con- 
nected, but  one  with  reversed  terminals.  This  method 
allows  a  secondary  neutral  also  to  be  brought  out. 

287.    Transformation   with   a   change   of    the   balance 
factor  of  the  system  is  possible  only  by  means  of  apparatus 


\ 

\ 

•/ 

/ 

y 

/  \ 

y 

2'   v    ' 

Fig.  208. 


able  to  store  energy,  since  the  difference  of  power  between 
primary  and  secondary  circuit  has  to  be  stored  at  the  time 
when  the  secondary  power  is  below  the  primary,  and  re- 
turned during  the  time  when  the  primary  power  is  below 


TRANSPORMATION  OF  POLYPHASE   SYSTEMS.       467 

the  secondary.  The  most  efficient  storing  device  of  electric 
energy  is  mechanical  momentum  in  revolving  machinery. 
It  has,  however,  the  disadvantage  of  requiring  attendance ; 
fairly  efficient  also  are  capacities  and  inductances,  but,  as  a 
rule,  have  the  disadvantage  not  to  give  constant  potential. 


468  ALTERNATING-CURRENT  PHENOMENA. 


CHAPTER    XXX. 

EFFICIENCY  OF  SYSTEMS. 

288.  In  electric  power  transmission  and  distribution, 
wherever  the  place  of  consumption  of  the  electric  energy 
is  distant  from  the  place  of  production,  the  conductors 
which  transfer  the  current  are  a  sufficiently  large  item  to 
require  consideration,  when  deciding  which  system  and 
•what  potential  is  to  be  used. 

In  general,  in  transmitting  a  given  amount  of  power  at  a 
given  loss  over  a  given  distance,  other  things  being  equal, 
the  amount  of  copper  required  in  the  conductors  is  inversely 
proportional  to  the  square  of  the  potential  used.  Since 
the  total  power  transmitted  is  proportional  to  the  product 
of  current  and  E.M.F.,  at  a  given  power,  the  current  will 
vary  inversely  proportional  to  the  E.M.F.,  and  therefore, 
since  the  loss  is  proportional  to  the  product  of  current- 
square  and  resistance,  to  give  the  same  loss  the  resistance 
must  vary  inversely  proportional  to  the  square  of  the  cur- 
rent,  that  is,  proportional  to  the  square  of  the  E.M.F.  ;  and 
since  the  amount  of  copper  is  inversely  proportional  to  the 
resistance,  other  things  being  equal,  the  amount  of  copper 
varies  inversely  proportional  to  the  square  of  the  E.M.F. 
used. 

This  holds  for  any  system. 

Therefore  to  compare  the  different  systems,  as  two-wire 
single-phase,  single-phase  three-wire,  three-phase  and  quar- 
ter-phase, equality  of  the  potential  must  be  assumed. 

Some  systems,  however,  as  for  instance,  the  Edison 
three-wire  system,  or  the  inverted  three-phase  system,  have 


EFFICIENCY  OF  SYSTEMS.  409 

different  potentials  in  the  different  circuits  constituting  the 
system,  and  thus  the  comparison  can  be  made  either  — 

1st.  On  the  basis  of  equality  of  the  maximum  potential 
difference  in  the  system  ;  or 

2d.  On  the  basis  of  the  minimum  potential  difference 
in  the  system,  or  the  potential  difference  per  circuit  or 
phase  of  the  system. 

In  low  potential  circuits,  as  secondary  networks,  where 
the  potential  is  not  limited  by  the  insulation  strain,  but  by 
the  potential  of  the  apparatus  connected  into  the  system, 
as  incandescent  lamps,  the  proper  basis  of  comparison  is 
equality  of  the  potential  per  branch  of  the  system,  or  per 
phase. 

On  the  other  hand,  in  long  distance  transmissions  where 
the  potential  is  not  restricted  by  any  consideration  of  ap- 
paratus suitable  for  a  certain  maximum  potential  only,  but 
where  the  limitation  of  potential  depends  upon  the  problem 
of  insulating  the  conductors  against  disruptive  discharge, 
the  proper  comparison  is  on  the  basis  of  equality  of  the 
maximum  difference  of  potential  in  the  system ;  that  is, 
•equal  maximum  dielectric  strain  on  the  insulation. 

The  same  consideration  holds  in  moderate  potential 
power  circuits,  in  considering  the  danger  to  life  from  live 
wires  entering  human  habitations. 

Thus  the  comparison  of  different  systems  of  long-dis- 
tance transmission  at  high  potential  or  power  distribution 
for  motors  is  to  be  made  on  the  basis  of  equality  of  the 
maximum  difference  of  potential  existing  in  the  system. 
The  comparison  of  low  potential  distribution  circuits  for 
lighting  on  the  basis  of  equality  of  the  minimum  difference 
of  potential  between  any  pair  of  wires  connected  to  the 
receiving  apparatus. 

289.  1st.  Comparison  on  the  basis  of  equality  of  the 
minimum  difference  of  potential,  in  low  potential  lighting 
circuits  : 


4TO  ALTERNATING-CURRENT  PHENOMENA. 

In  the  single-phase  alternating-current  circuit,  if  e  — 
E.M.F.,  i  =  current,  r—  resistance  per  line,  the  total  power 
is  =  ei,  the  loss  of  power  2z'V. 

Using,  however,  a  three-wire  system,  the  potential  be- 
tween outside  wires  and  neutral  being  given  =  e,  the 
potential  between  the  outside  wires  is  ==  2  e,  that  is,  the  dis- 
tribution takes  place  at  twice  the  potential,  or  only  -'•  the 
copper  is  needed  to  transmit  the  same  power  at  the  same 
loss,  if,  as  it  is  theoretically  possible,  the  neutral  wire  has 
no  cross-section.  If  therefore  the  neutral  wire  is  made  of 
the  same  cross-section  with  each  of  the  outside  wires,  |  of 
the  copper  of  the  two- wire  system  is  needed  ;  if  the  neutral 
wire  is  £  the  cross-section  of  each  of  the  outside  wires,  T%  of 
the  copper  is  needed.  Obviously,  a  single-phase  five-wire 
system  will  be  a  system  of  distribution  at  the  potential  4  e, 
and  therefore  require  only  TV  °f  the  copper  of  the  single- 
phase  system  in  the  outside  wires ;  and  if  each  of  the  three 
neutral  wires  is  of  i  the  cross-section  of  the  outside  wires, 
/?  =  10.93  per  cent  of  the  copper. 

Coming  now  to  the  three-phase  system  with  the  poten- 
tial e  between  the  lines  as  delta  potential,  if  i  =  the  current 
per  line  or  Y  current,  the  current  from  line  to  line  or  delta 
current  =  ^  /  VB  ;  and  since  three  branches  are  used,  the 
total  power  is  3  e  i\  /  V3  ==  e  z'x  V3.  Hence  if  the  same 
power  has  to  be  transmitted  by  the  three-phase  system  as 
with  the  single-phase  system,  the  three-phase  line  current 
must  be  z'i  =  i  /  V3  where  i  —  single-phase  current,  r  = 
single-phase  resistance  per  line,  at  equal  power  and  loss; 
hence  if  1\  =  resistance  of  each  of  the  three  wires,  the  loss 
per  wire  is  i?  rt  =  iz  rt  /.3,  and  the  total  loss  is  z2 1\,  while  in 
the  single-phase  system  it  is  2  t*r.  Hence,  to  get  the  same 
loss,  it  must  be :  rv  =  2  r,  that  is,  each  of  the  three  three- 
phase  lines  has  twice  the  resistance  —  that  is,  half  the  cop- 
per of  each  of  the  two  single-phase  lines  ;  or  in  other  words, 
the  three-phase  system  requires  three-fourths  of  the  copper 
of  the  single-phase  system  of  the  same  potential. 


EFFICIENCY  OF  SYSTEMS.  471 

Introducing,  however,  a  fourth  or  neutral  wire  into  the 
three-phase  system,  and  connecting  the  lamps  between  the 
neutral  wire  and  the  three  outside  wires  —  that  is,  in  Y  con- 
nection—  the  potential  between  the  outside  wires  or  delta 
potential  will  be  =  e  X  V3,  since  the  Y  potential  =  e,  and 
the  potential  of  the  system  is  raised  thereby  from  e  to 
e  V3  ;  that  is,  only  J  as  much  copper  is  required  in  the  out- 
side wires  as  before  —  that  is  \  as  much  copper  as  in  the 
single-phase  two-wire  system.  Making  the  neutral  of  the 
same  cross-section  as  the  outside  wires,  requires  \  more 
copper,  or  \  =  33.3  per  cent  of  the  copper  of  the  single- 
phase  system  ;  making  the  neutral  of  half  cross-section, 
requires  \  more,  or  ^  =  29.17  per  cent  of  the  copper  of 
the  single-phase  system.  The  system,  however,  now  is  a 
four-wire  system. 

The  independent  quarter-phase  system  with  four  wires 
is  identical  in  efficiency  to  the  two-wire  single-phase  sys- 
tem, since  it  is  nothing  but  two  independent  single-phase 
systems  in  quadrature. 

The  four-wire  quarter-phase  system  can  be  used  as  two 
independent  Edison  three-wire  systems  also,  deriving  there- 
from the  same  saving  by  doubling  the  potential  between 
the  outside  wires,  and  has  in  this  case  the  advantage,  that 
by  interlinkage,  the  same  neutral  wire  can  be  used  for  both 
phases,  and  thus  one  of  the  neutral  wires  saved. 

In  this  case  the  quarter-phase  system  with  common  neu- 
tral of  full  cross-section  requires  -fo  =  31.25  per  cent,  the 
quarter-phase  system  with  common  neutral  of  one-half  cross- 
section  requires  ^  =  28.125  per  cent,  of  the  copper  of  the 
two-wire  single-phase  system. 

In  this  case,  however,  the  system  is  a  five-wire  system, 
and  as  such  far  inferior  to  the  five-wire  single-phase  system. 

Coming  now  to  the  quarter-phase  system  with  common 
return  and  potential  e  per  branch,  denoting  the  current  in 
the  outside  wires  by  z'2,  the  current  in  the  central  wire  is 
*a  V2  ;  and  if  the  same  current  density  is  chosen  for  all 


472  ALTERNATING-CURRENT  PHENOMENA. 

three  wires,  as  the  condition  of  maximum  efficiency,  and 
the  resistance  of  each  outside  wire  denoted  by  rz,  the  re- 
sistance of  the  central  wire  =  r2/V2,  and  the  loss  of  power 
per  outside  wire  is  z'22  r2 ,  in  the  central  wire  2  z'22  r2  /  V2 
=  z'22  r2  V2 ;  hence  the  total  loss  of  power  is  2  z'22  r2  +  z'22  r2 
V2  =  z'22  r2  (2  -f  V2).  The  power  transmitted  per  branch 
is  z'2  ^,  hence  the  total  power  2  z'2  e.  To  transmit  the  same 
power  as  by  a  single-phase  system  of  power,  e  z,  it  must 

be  z2  =  z'/2;  hence  the  loss,  *2;a(2  +  ^ .  Since  this 
loss  shall  be  the  same  as  the  loss  2z'2r  in  the  single- 
phase  system,  it  must  be  2  r  =  - —  —  r2 ,  or  r2  = ~ . . 

2  -}-  V  2 

°  4-   V^ 
Therefore  each  of  the  outside  wires  must  be  —         —  times 

o 

as  large  as  each  single-phase  wire,  the  central  wire  V2 
times  larger ;  hence  the  copper  required  for  the  quarter- 
phase  system  with  common  return  bears  to  the  copper 
required  for  the  single-phase  system  the  relation  : 

2  (2  +  V2)       (2  +  V5)  V2  .   9         3  +  2V2 

^~  ~T~  ~T~~ 

per  cent  of  the  copper  of  the  single-phase  system. 

Hence  the  quarter-phase  system  with  common  return 
saves  2  per  cent  more  copper  than  the  three-phase  system, 
but  is  inferior  to  the  single-phase  three-wire  system. 

The  inverted  three-phase  system,  consisting  of  two 
E.M.Fs.  e  at  60°  displacement,  and  three  equal  currents 
/8  in  the  three  lines  of  equal  resistance  r3,  gives  the  out- 
put 2^z'3,  that  is,  compared  with  the  single-phase  system, 
/8  =  z'/2.  The  loss  in  the  three  lines  is  3  z'32  r3  =  |  z2  rs. 
Hence,  to  give  the  same  loss  2  z'2  r  as  the  single-phase  sys- 
tem, it  must  be  rs  =  f  r,  that  is,  each  of  the  three  wires 
must  have  f  of  the  copper  cross-section  of  the  wire  in  the 
two-wire  single-phase  system  ;  or  in  other  words,  the  in- 
verted three-phase  system  requires  ^  of  the  copper  of  the 
two-wire  single-phase  system. 


EFFICIENCY  OF  SYSTEMS. 


473 


We  get  thus  the  result, 

If  a  given  power  has  to  be  transmitted  at  a  given  loss, 
and  a  given  minimum  potential,  as  for  instance  110  volts 
for  lighting,  the  amount  of  copper  necessary  is  : 

2  WIRES  :    Single-phase  system,  100.0 

3  WIRES  :    Edison   three-wire    single-phase    sys- 

tem, neutral  full  section,  37.5 
Edison    three-wire    single-phase  sys- 
tem, neutral  half-section,  31.25 
Inverted  three-phase  system,  56.25 
Quarter-phase  system  with  common 

return,  72.9 

Three-phase  system,  75.0 

4  WIRES  :    Three-phase,  with   neutral  wire    full 

section,  33.3 

Three-phase,  with  neutral  wire    half- 
section,  29.17 
Independent  quarter-phase  system,      100.0 

5  WIRES  :    Edison  five-wire,  single-phase  system, 

full  neutral,  15.625 

Edison  five-wire,  single-phase  system, 

half-neutral,  10.93 

Four-wire,    quarter-phase,  with  com- 
mon neutral  full  section,  31.25 
Four-wire,    quarter-phase,  with  com- 
mon neutral   half-section,  28.125 


We  see  herefrom,  that  in  distribution  for  lighting  —  that 
is,  with  the  same  minimum  potential,  and  with  the  same 
number  of  wires  —  the  single-phase  system  is  superior  to 
any  polyphase  system. 

The  continuous-current  system  is  equivalent  in  this' 
comparison  to  the  single-phase  alternating-current  system 
of  the  same  effective  potential,  since  the  comparison  is 
made  on  the  basis  of  effective  potential,  and  the  power 
depends  upon  the  effective  potential  also. 


474  AL  TERNA  TING-CURRENT  PHENOMENA. 

290.  Comparison  on  the  Basis  of  Equality  of  the  Maximum 
Difference  of  Potential  in  the  System,  in  Long-  Distance 
Transmission,  Power  Distribution,  etc. 

Wherever  the  potential  is  so  high  as  to  bring  the  ques- 
tion of  the  strain  on  the  insulation  into  consideration,  or  in 
other  cases,  to  approach  the  danger  limit  to  life,  the  proper 
comparison  of  different  systems  is  on  the  basis  of  equality 
of  maximum  potential  in  the  system. 

Hence  in  this  case,  since  the  maximum  potential  is 
fixed,  nothing  is  gained  by  three-  or  five-wire  Edison  sys- 
tems. Thus,  such  systems  do  not  come  into  consideration. 

The  comparison  of  the  three-phase  system  with  the 
single-phase  system  remains  the  same,  since  the  three- 
phase  system  has  the  same  maximum  as  minimum  poten- 
tial ;  that  is  : 

The  three-phase  system  requires  three-fourths  of  the 
copper  of  the  single-phase  system  to  transmit  the  same 
power  at  the  same  loss  over  the  same  distance. 

The  four-wire  quarter-phase  system  requires  the  same 
amount  of  copper  as  the  single-phase  system,  since  it  con- 
sists of  two  single-phase  systems. 

In  a  quarter-phase  system  with  common  return,  the 
potential  between  the  outside  wire  is  V2  times  the  poten- 
tial per  branch,  hence  to  get  the  same  maximum  strain  on 
the  insulation  —  that  is,  the  same  potential  e  between  the 
outside  wires  as  -in  the  single-phase  system  —  the  potential 
per  branch  will  be  ej  V2,  hence  the  current  z'4  =  t/  V2,  if  i 
equals  the  current  of  the  single-phase  system  of  equal 
power,  and  t\  V2  =  i  will  be  the  current  in  the  central 
wire. 

Hence,  if  r±  =  resistance  per  outside  wire,  r±  /  V2  = 
resistance  of  central  wire,  and  the  total  loss  in  the  sys- 
tem is  : 


,  (2  +  V2)  = 


EFFICIENCY  OF  SYSTEMS.  475 

Since  in  the  single-phase  system,  the  loss  =  2  i  2  r,  it  is  : 


2  +  ~v/2 
That  is,  each  of  the  outside  wires  has  to  contain  —  —  -  - 

4 
times  as  much  copper  as  each  of  the   single-phase  wires. 

2  x  V2    /- 
The    central    wires    have    to    contain  -  -  V  2   times  as 

^  (^  -4-  ~v/2^ 
much  copper  ;  hence  the  total  system  contains 


2  +V2 
—  T  -  V2  times  as  much  copper  as  each  of  the  single- 

3  +  2  ~\/2 

phase  wires  ;    that  is,  -  —  times  the  copper  of  the 

4 

single-phase  system. 
Or,  in  other  words, 
A  quarter-phase  system  with  common  return  requires 

3  +  2  A/2 

—  ==  1.457  times  as  much  copper  as  a  single-phase 

system  of  the  same  maximum  potential,  same  power,  and 
same  loss. 

Since  the  comparison  is  made  on  the  basis  of  equal 
maximum  potential,  and  the  maximum  potential  of  alter- 
nating system  is  A/2  times  that  of  a  continuous-current 
circuit  of  equal  effective  potential,  the  alternating  circuit 
of  effective  potential  e  compares  with  the  continuous-cur- 
rent circuit  of  potential  e  A/2,  which  latter  requires  only 
half  the  copper  of  the  alternating  system. 

This  comparison  of  the  alternating  with  the  continuous- 
current  system  is  not  proper  however,  since  the  continuous- 
current  potential  introduces,  besides  the  electrostatic  strain, 
an  electrolytic  strain  on  the  dielectric  which  does  not  exist 
in  the  alternating  system,  and  thus  makes  the  action  of  the 
continuous-current  potential  on  the  insulation  more  severe 
than  that  of  an  equal  alternating  potential.  Besides,  self- 
induction  having  no  effect  on  a  steady  current,  continuous 
current  circuits  as  a  rule  have  a  self-induction  far  in  excess 


476  ALTERNATING-CURRENT  PHENOMENA. 

of  any  alternating  circuit.  During  changes  of  current,  as 
make  and  break,  and  changes  of  load,  especially  rapid 
changes,  there  are  consequently  induced  in  these  circuits 
E.M.F.'s  far  exceeding  their  normal  potentials.  At  the 
voltages  which  came  under  consideration,  the  continuous 
current  is  excluded  to  begin  with. 

Thus  we  get : 

If  a  given  power  is  to  be  transmitted  at  a  given  loss, 
and  a  given  maximum  difference  of  potential  in  the  system, 
that  is,  with  the  same  strain  on  the  insulation,  the  amount 
of  copper  required  is  : 

2  WIRES  :  Single-phase  system,  100.0 

[Continuous-current  system,  50.0] 

3  WIRES  :  Three-phase  system,  75.0 

Quarter-phase  system,  with  common  return,  145.7 

4  WIRES  :  Independent  Quarter-phase  system,  100.0 

Hence  the  quarter-phase  system  with  common  return  is 
practically  excluded  from  long-distance  transmission. 

291 .  In  a  different  way  the  same  comparative  results 
between  single-phase,  three-phase,  and  quarter-phase  sys- 
tems can  be  derived  by  resolving  the  systems  into  their 
single-phase  branches. 

The  three-phase  system  of  E.M.F.  e  between  the  lines 
can  be  considered  as  consisting  of  three  single-phase  cir- 
cuits of  E.M.F.  ^/V3,  and  no  return.  The  single-phase 
system  of  E.M.F.  e  between  lines  as  consisting  of  two 
single-phase  circuits  of  E.M.F.  <?/2  and  no  return.  Thus, 
the  relative  amount  of  copper  in  the  two  systems  being 
inversely  proportional  to  the  square  of  E.M.F.,  bears  the 
relation  ( V3  /  e)2  :  (2  /  ef  =  3  :  4  ;  that  is,  the  three-phase 
system  requires  75  per  cent  of  the  copper  of  the  single- 
phase  system. 

The  quarter-phase  system  with  four  equal  wires  requires 
the  same  copper  as  the  single-phase  system,  since  it  consists 


EFFICIENCY  OF  SYSTEMS.  477 

of  two  single-phase  circuits.  Replacing  two  of  the  four 
quarter-phase  wires  by  one  wire  of  the  same  cross-section 
as  each  of  the  wires  replaced  thereby,  the  current  in  this 
wire  is  V2  times  as  large  as  in  the  other  wires,  hence,  the 
loss  twice  as  large  —  that  is,  the  same  as  in  the  two  wires 
replaced  by  this  common  wire,  or  the  total  loss  is  not 
changed  —  while  25  per  cent  of  the  copper  is  saved,  and 
the  system  requires  only  75  per  cent  of  the  copper  of  the 
single-phase  system,  but  produces  V2  times  as  high  a 
potential  between  the  outside  wires.  Hence,  to  give  the 
same  maximum  potential,  the  E.M.Fs.  of  the  system  have 
to  be  reduced  by  V2,  that  is,  the  amount  of  copper  doubled, 
and  thus  the  quarter-phase  system  with  common  return  of 
the  same  cross-section  as  the  outside  wires  requires  150 
per  cent  of  the  copper  of  the  single-phase  system.  In  this 
case,  however,  the  current  density  in  the  middle  wire  is 
higher,  thus  the  copper  not  used  most  economical,  and 
transferring  a  part  of  the  copper  from  the  outside  wire  to 
the  middle  wire,  to  bring  all  three  wires  to  the  same  current 
density,  reduces  the  loss,  and  thereby  reduces  the  amount 
of  copper  at  a  given  loss,  to  145.7  per  cent  of  that  of  a 
single-phase  system. 


478  ALTERNATING-CURRENT  PHENOMENA. 


CHAPTER   XXXI. 

THREE-PHASE    SYSTEM. 

292.  With  equal  load  of  the  same  phase  displacement 
in  all  three  branches,  the  symmetrical  three-phase  system 
offers  no  special  features  over  those  of  three  equally  loaded 
single-phase  systems,  and  can  be  treated  as  such  ;  since  the 
mutual  reactions  between  the  three  phases  balance  at  equal 
distribution  of  load,  that  is,  since  each  phase  is  acted  upon 
by  the  preceding  phase  in  an  equal  but  opposite  manner 
as  by  the  following  phase. 

With  unequal  distribution  of  load  between  the  different 
branches,  the  voltages  and  phase  differences  become  more  or 
less  unequal.  These  unbalancing  effects  are  obviously  maxi- 
mum, if  some  of  the  phases  are  fully  loaded,  others  unloaded, 

Let: 

E  —  E.M.F.  between  branches  1  and  2  of  a  three-phaser. 
Then: 

«  E  =  E.M.F.  between  2  and  3, 
(*£=  E.M.F.  between  3  and  1, 


where,  e=  ^1=  ~        - 

Let 

ZD  Z2,  Zs  =  impedances  of  the  lines  issuing  from  genera- 

tor terminals  1,  2,  3, 
and   Yl}  Y2,  Ys  =  admittances   of   the   consumer  circuits   con- 

nected between  lines  2  and  3,  3  and  1,  1  and  2. 
Jf  then, 

ID  It,  /8,  are  the  currents  issuing  from  the  generator  termi- 
nals into  the  lines,  it  is, 

/I  +   /2  +   /3    =    0.  (1) 


THREE-PHASE   SYSTEM.  479 

If      //,  72',  7/  =  currents  flowing  through  the  admittances    Y1, 
F2,  F3,  from  2  to  3,  3  to  1,  1  to  2,  it  is, 

/!  =  /,'-/,',    or,   /1  +  /2'_/3'  =  Ol 
>,->/-/.',    or,   /2  +  /3'-7/  =  o[  (2) 

>3  =  //->/,    or,   /3  +  >1/-//  =  OJ 

These  three  equations  (2)  added,  give  (1)  as  dependent 
equation. 

At  the  ends  of  the  lines  1,  2,  3,  it  is  : 


(3) 
Il  +  ztIt)  • 

the  differences  of  potential,  and 

ti 

(4) 


the  currents  in  the  receiver  circuits. 

These  nine  equations   (2),   (3),   (4),  determine  the   nine 
quantities  :  flt  72,  /3,  //,  7a',  73',  ^',  Ti^  £&• 

Equations  (4)  substituted  in  (2)  give  : 


(5) 


These  equations  (5)  substituted  in  (3),  and  transposed, 
give, 

since      £l  =  c  E 

Ez  =  £  E  \  as  E.M.Fs.  at  the  generator  terminals. 


480  AL  TERNA  TING-CURRENT  PHENOMENA. 

as   three   linear    equations  with    the    three   quantities  2T/, 

Substituting  the  abbreviations  : 

a      I      \7   7       I      I/"   7  \  I/"    7  ~\7    7       i 

~T  *  1^2  ~T  *1^3)>          -tZ^S)          •*8^'2    I 

7  V   7  /1_1_V7_1_V7N>/ 

^zt      y 2-^D      —  V*1  ~r  -^s^i  T  *»^V  / 


A 


c,      F2Z3,      F3Z2 

a,      -  (1  +  ^^3  + 

,      Y,Zlt      -(1  +  F3Z1+F3Z2) 

-  (1  +  Y,Z2  +  FiZ,),     c,      F3Z2 
F.Z3,     c2,      YtZ, 
Y.Z,,     1,      -  (1  +  F3ZX  +  F3Z2) 

(i  +  ^iz.  +  yiz,),    F2z3,    £ 


A  =  /  FIZS,    -  (i 

FaZ2,      F2ZX, 


it  is: 


D 


72  =    i 


__  F2Z>2- 


hence, 


(8) 


(9) 


(10) 


(11) 


THREE-PHASE  SYSTEM. 

293.    SPECIAL    CASES. 

A.    Balanced  System 

Y,  =  F2  =  F8  =  F 
Z,  =  Z2  =  Z3  =  Z. 

Substituting  this  in  (6),  and  transposing  : 


481 


c  E 


£s  =  £ 


EI  = 


3FZ 


1  +  3FZ 


1  +  3YZ 
EY 


1  +  3KZJ 


3FZ 


3FZ 


3  YZ 


(12) 


The  equations  of  the  symmetrical  balanced  three-phase 
system. 

B.    One  circuit  loaded,  two  unloaded: 

F!  =  F2  =  0,    F8  =  F 
Zj  =  Z2  =  Z3  =  Z. 

Substituted  in  equations  (6) : 

=      (  unloaded  branches. 
E  —  E3'(l  +  2  FZ)  =  0,  loaded  branch. 


hence  : 
r./ 

, 


2KZ 


2FZ 


1  +  2  FZ 


unloaded ; 


loaded  ; 


all  three 

KM.F.'s 

unequal,  and  (13) 
of  unequal 
phase  angles. 


482 


AL  TERNA  TING-CURRENT  PHENOMENA. 


(13) 


(13) 


C.    Two  circuits  loaded,  one  tinloaded. 

F!  =  F2  =  F,      F8  =  0, 
Zt  =  Z2  =  Z3  =  Z. 

Substituting  this  in  equations  (6),  it  is  : 

e  E  —  E{  (1  +  2  FZ)  +  .£/  FZ  =  0) 
£E  —  El  (1  +  2  FZ)  +  E{  FZ  =  0  J 

E  —  £s'  +  (,£,'  +  ^2')  FZ  =  0     unloaded  branch, 
or,  since : 

E  —  Ez''—  EZ'Y2 :'=  0, 

E1  =       ? 

\  +  FZ 

thus: 


1  +  4  FZ  +  3  F2Z2 


1  +  4  FZ  +  3  F2Z2 
E 

I+'FZ 


loaded  branches. 


unloaded  branch. 


(14) 


As  seen,  with  unsymmetrical  distribution  of  load,  all 
three  branches  become  more  or  less  unequal,  and  the  phase 
displacement  between  them  unequal  also. 


QUARTER-PHASE  SYSTEM.  483 


CHAPTER    XXXII. 

QUARTER-PHASE    SYSTEM. 

294.  In  a  three-wire  quarter-phase  system,  or  quarter- 
phase  system  with  common  return  wire  of  both  phases,  let 
the  two  outside  terminals  and  wires  be  denoted  by  1  and  2> 
the  middle  wire  or  common  return  by  0. 

It  is  then  : 

EI  =  E  =  E.M.F.  between  0  and  1  in  the  generator. 
Ez=jE  =  E.M.F.  between  0  and  2  in  the  generator. 

Let: 

./i  and  72  =  currents  in  1  and  in  2, 
70  =  current  in  0, 

Z-L  and  Zz  =  impedances  of  lines  1  and  2, 
Z0  =  impedance  of  line  0. 

Yl  and  Y2  =  admittances  of  circuits  0  to  1,  and  0  to  2, 
//  and  //=  currents  in  circuits  0  to  1,  and  0  to  2, 
Eia.-ndE2'=  potential  differences  at  circuit  0  to  1,  and 
0  to  2. 

it  is  then,  7,  -f  78  +  70  =  0  )  «v 

or,  I0  =-(/;  +  72)  j 

that  is,  70  is  common  return  of  7:  and  72. 

Further,  we  have, 


El  =JE  -  72  Z0  +    0Z0  =jE  -  72  (Z2  +  Z0)  -  A 

and 

A  =  K,  E{ 

(3) 


484  AL  TERNA  TING-CURRENT  PHENOMENA. 

Substituting  (3)  in  (2) ;  and  expanding : 
*•/  -  *•  _  l  +  F2Z2  +  F2Z0(l-y) _ 

'.  (4) 

•    2  •     /1_l_VX_l_V7'W'l_l_V7_l_V5^          V  V    '7  2 

\*-   i   *  1^0  "T"  *\**\)\~   i   * i **•  T  * i^ij  —  *i *J ^o 

Hence,  the  two  E.M.Fs.  at  the  end  of  the  line  are  un- 
equal in  magnitude,  and  not  in  quadrature  any  more. 

295.    SPECIAL  CASES  : 

A.    Balanced  System. 

Z0  =  Z  /  V2  ; 
F,  =  F2  =  F 

Substituting  these  values  in  (4),  gives  : 

i  + 1  + V2-yrz 

'    1  +  V2  (1  +  V2)  FZ  +  (1  +  V2)  F2Z; 


_  E    1  +  (1.707  -  .707/)  FZ 
•    1  +  3.414  FZ  +  2.414  F2Z2 


(5) 


V2 
~J  •   1  +  V2  (1  +  V2)  FZ  +  (1  +  V2)  F2Z2 

_  . ^     1  +  (1.707  +  .707.;)  FZ 
'    1  +  3.414  FZ  +  2.414  F2Z2 

Hence,  the  balanced  quarter-phase  system  with  common 
return  is  unbalanced  with  regard  to  voltage  and  phase  rela- 
tion, or  in  other  words,  even  if  in  a  quarter-phase  system  with 
common  return  both  branches  or  phases  are  loaded  equally, 
with  a  load  of  the  same  phase  displacement,  nevertheless 
the  system  becomes  unbalanced,  and  the  two  E.M.Fs.  at 
the  end  of  the  line  are  neither  equal  in  magnitude,  nor  in 
quadrature  with  each  other. 


QUARTER-PHASE  SYSTEM. 
B.    One  branch  loaded,  one  unloaded. 


485 


a.) 
b.) 


Substituting  these  values  in  (4),  gives  : 

i  +  V2  —  y 


b.} 


l  +  FZ 


a.)  £1  =  E 


V2 


1  +  V2 
V2 
j 


2.414  + 


1.414 
YZ 


*+'*f 

=  /^l4-1.707FZ 


1+^1^ 


•   1  +  1.707  FZ 

-t       I      ^/O 

1  +  F2 


V2 


, 


FZ 


1  + 


+ 


V2 


2.414  + 


1.414 
FZ 


(6) 


486  AL  TERNA  TING-CURRENT  PHENOMENA. 

These  two  E.M.Fs.  are  unequal,  and  not  in  quadrature 
with  each  other. 

But  the  values  in  case  a.)  are  different  from  the  values 
in  case  b.}. 

That  means  : 

The  two  phases  of  a  three-wire  quarter-phase  system 
are  unsymmetrical,  and  the  leading  phase  1  reacts  upon 
the  lagging  phase  2  in  a  different  manner  than  2  reacts 
upon  1. 

It  is  thus  undesirable  to  use  a  three-wire  quarter-phase 
system,  except  in  cases  where  the  line  impedances  Z  are 
negligible. 

In  all  other  cases,  the  four-wire  quarter-phase  system 
is  preferable,  which  essentially  consists  of  two  independent 
single-phase  circuits,  and  is  treated  as  such. 

Obviously,  even  in  such  an  independent  quarter-phase 
system,  at  unequal  distribution  of  load,  unbalancing  effects 
may  take  place. 

If  one  of  the  branches  or  phases  is  loaded  differently 
from  the  other,  the  drop  of  voltage  and  the  shift  of  the 
phase  will  be  different  from  that  in  the  other  branch  ;  and 
thus  the  E.M.Fs.  at  the  end  of  the  lines  will  be  neither 
equal  in  magnitude,  nor  in  quadrature  with  each  other. 

With  both  branches  however  loaded  equally,  the  system 
remains  balanced  in  voltage  and  phase,  just  like  the  three- 
phase  system  under  the  same  conditions. 

Thus  the  four-wire  quarter-phase  system  and  the  three- 
phase  system  are  balanced  with  regard  to  voltage  and  phase 
at  equal  distribution  of  load,  but  are  liable  to  become  un- 
balanced at  unequal  distribution  of  load ;  the  three-wire 
quarter-phase  system  is  unbalanced  in  voltage  and  phase, 
even  at  equal  distribution  of  load. 


APPENDICES. 


APPENDIX    I. 


ALGEBRA   OF   COMPLEX    IMAGINARY 
QUANTITIES. 

INTRODUCTION. 

296.  The  system  of   numbers,  of  which   the   science 
of  algebra  treats,   finds  its   ultimate  origin  in  experience. 
Directly  derived   from    experience,  however,  are  only  the 
absolute  integral  numbers  ;  fractions,  for  instance,  are  not 
directly  derived  from  experience,  but  are  abstractions  ex- 
pressing relations  between  different  classes  of  quantities. 
Thus,  for  instance,  if  a  quantity  is  divided  in  two  parts, 
from  one  quantity  two  quantities  are  derived,  and  denoting 
these  latter  as  halves  expresses  a  relation,  namely,  that  two 
of  the  new  kinds  of  quantities  are  derived  from,  or  can  be 
combined  to  one  of  the  old  quantities. 

297.  Directly  derived  from  experience  is  the  operation 
of  counting  or  of  numeration. 

a,     a  +  1,     a  +  2,     a  +  3  .  .  .   . 
Counting  by  a  given  number  of  integers  : 


b  integers 
introduces  the  operation  of  addition,  as  multiple  counting  : 

a  +  b  =  c. 
It  is,  a  +  b  =  b  +  a, 


490  APPENDIX  7. 

that  is,  the  terms  of  addition,  or  addenda,  are  interchange- 
able. 

Multiple  addition  of  the  same  terms  : 

a  -+-  a  -\-  a  -+-  .  .  .  +  a  =  c 

b  equal  numbers 
introduces  the  operation  of  multiplication  : 

a  x  b  =  c. 
It  is,  a  X  b  =  b  X  a, 

that  is,  the  terms  of  multiplication,  or  factors,  are  inter- 
changeable. 

Multiple  multiplication  of  the  same  factors  : 

aX  aX  aX  .  .  •  X  a  =  c 

b  equal  numbers 
introduces  the  operation  of  involution  : 


Since  ab  is  not  equal  to  #", 

the  terms  of  involution  are  not  interchangeable. 

298.  The  reverse  operation  of  addition  introduces  the 
operation  of  subtraction  : 

If  a  +  6  =  f, 

it  is  c  —  b  =  a. 

This  operation  cannot  be  carried  out  in  the  system  of 
absolute  numbers,  if : 

b>  c. 

Thus,  to  make  it  possible  to  carry  out  the  operation  of 
subtraction  under  any  circumstances,  the  system  of  abso- 
lute numbers  has  to  be  expanded  by  the  introduction  of 
the  negative  number: 

_  «  =  (_  1)  X  «, 
.where  (-  1) 

is  the  negative  unit. 

Thereby  the  system  of  numbers  is   subdivided  in  the 


COMPLEX  IMAGINARY  QUANTITIES.  491 

positive  and  negative  numbers,  and  the  operation  of  sub- 
traction possible  for  all  values  of  subtrahend  and  minuend. 
From  the  definition  of  addition  as  multiple  numeration,  and 
subtraction  as  its  inverse  operation,  it  follows  : 

c  -  (-  b)  =  c  +  b, 
thus:  (-l)X  (-!)  =  !; 

that  is,  the  negative  unit  is  defined  by,  (—I)2  =  1. 

299.  The  reverse  operation  of  multiplication  introduces 
the  operation  of  division  : 

If  a  X  b  =  c,  then  -  =  a. 

b 

In  the  system  of  integral  numbers  this  operation  can 
only  be  carried  out,  if  b  is  a  factor  of  c. 

To  make  it  possible  to  carry  out  the  operation  of  division 
under  any  circumstances,  the  system  of  integral  numbers 
has  to  be  expanded  by  the  introduction  of  infraction: 


:©. 

where  -  is  the  integer  fraction,  and  is  defined  by  : 


T-    x  b  =  1. 


300.    The  reverse  operation  of  involution  introduces  two 
new  operations,  since  in  the  involution  : 


the  quantities  a  and  b  are  not  reversible. 

Thus  V^  =  <z,  the  evolution, 

=  b,  the  logarithmation. 


The  operation  of  evolution  of  terms  c,  which  are  not 
•complete  powers,  makes  a  further  expansion  of  the  system 


492  APPENDIX  I. 

of  numbers  necessary,  by  the  introduction  of  the  irrational 
number  (endless  decimal  fraction),  as  for  instance  : 

V2  =  1.414213. 

301.    The  operation  of  evolution  of  negative  quantities 
c  with  even  exponents  b,  as  for  instance 

2/  - 

makes  a  further  expansion  of  the  system  of  numbers  neces- 
sary, by  the  introduction  of  the  imaginary  unit. 

-V^l 
Thus  -x/^  =  -v/^T  x  •#*. 

where  :    V—  1  is  denoted  by/. 

Thus,  the  imaginary  unity  is  defined  by  : 

f  =  _  1. 

By  addition  and  subtraction  of  real  and  imaginary  units, 
compound  numbers  are  derived  of  the  form  : 


which  are  denoted  as  complex  imaginary  mimbers. 

No  further  system  of  numbers  is  introduced  by  the 
operation  of  evolution. 

The  operation  of  logarithmation  introduces  the  irrational 
and  imaginary  and  complex  imaginary  numbers  also,  but 
no  further  system  of  numbers. 

302.  Thus,  starting  from  the  absolute  integral  num- 
bers of  experience,  by  the  two  conditions  : 

1st.  Possibility  of  carrying  out  the  algebraic  operations 
and  their  reverse  operations  under  all  conditions, 

2d.    Permanence  of  the  laws  of  calculation, 
the  expansion  of  the  system  of  numbers  has  become  neces- 
sary, into 

Positive  and  negative  numbers, 

Integral  numbers  and  fractions, 

Rational  and  irrational  numbers, 


COMPLEX  IMAGINARY  QUANTITIES.  493 

Real  and  imaginary  numbers  and  complex  imaginary 
numbers. 

Therewith  closes  the  field  of  algebra,  and  all  the  alge- 
braic operations  and  their  reverse  operations  can  be  carried 
out  irrespective  of  the  values  of  terms  entering  the  opera- 
tion. 

Thus  within  the  range  of  algebra  no  further  extension 
of  the  system  of  numbers  is  necessary  or  possible,  and  the 
most  general  number  is 

a  +  jb. 

where  a  and  b  can  be   integers  or  fractions,   positive  or 
negative,  rational  or  irrational. 

ALGEBRAIC  OPERATIONS  WITH  COMPLEX  IMAGINARY 
QUANTITIES. 

303.    Definition  of  imaginary  unit: 

f2  =  -  1. 
Complex  imaginary  number: 


Substituting  : 

a  =  r  cos  (3 
b  =  r  sin  (3, 
it  is  A  =  r  (cos  /3  -f  /  sin  /?), 

where  r  =     a2  --    \ 


a 

r  =  vector, 
/3  =  amplitude  of  complex  imaginary  number  A. 

Substituting  : 

eJft  4-  c-JP 

H 


cos 


sin/?  = 


494  APPENDIX  I. 

it  is  A  =  reJP, 

where  c  =  lim  (l  +  -}"=  yJT  _  1 
„=»  V         n)          o~lx2X3x 

is  the  basis  of  the  natural  logarithms. 
Conjugate  numbers  : 

a  -\-  j  b  =  r  (cos  ft  -\-  j  sin  ft)  =  reJ'P 
•and  a  —  jb  =  r  (cos  [—/?]+>  sin  [—  /?]) 
it  is 


Associate  numbers: 
a  +  jb  =  r  (cos  ft  +/  sin  /3)  = 


and  b  + 

ja  =  r  (  cos  1  ^  —  (3\  -f  j  sin  \7- 

it  is 

(a+jb)(b+ja)=j(a*+P) 

If 

a+jb  =  a'  +jb', 

it  is 

a  =  af 

If 

a  +J/=  0  ; 

it  is 

a  =  0, 

304.    Addition  and  Subtraction  : 


Multiplication  : 

(a  +jb)  (a'  +jb')  =  (aa1  -  b  b')  +j(ab'  +  b  a') 
or     r  (cos  ^3  +  /  sin  ft)  X  r'  (cos  /?  +  /  sin  ftf)  =  r  r'  (cos  [£  -p 

^]+ysin[/3  +  ^]); 
or     re  J*  X  r'^'07  =  rr'ef&  +  M. 

Division  : 

Expansion  of  complex  imaginary  fraction,  for  rationaliza- 
tion of  denominator  or  numerator,  by  multiplication  with 
the  conjugate  quantity  : 


COMPLEX  IMAGINARY  QUANTITIES.  495*" 

a+jb  =   (a+jb}(a'  -jb'}   =  (aar+  bb'}  +j  (b  a' -  ab'} 
-jb'}  ,         *"  +  *" 


(a!  -f  j  b'}  (a  —  jb}       (a  a'  +  b  b'}  +j(ab'  —  b  a')  ' 


or,  _  r  ^_p  ^       _  ^    . 

r' 


or> 

r 


involution  : 

(a  +jbY  =  {r  (cos 


evolution  : 


-v/^-  (cos  /8  +  y  sin 


305.    Roots  of  the  Unit  : 
=+l,     -1; 


</I=+i,    -i,    +y,    -y; 

'  +i+y   +i-y   -i +y 


V2  V2  V2 

-i-y 

V2      ' 


306.    Rotation  : 

In  the  complex  imaginary  plane, 
multiplication  with 

9  *  2-n- 

VI  =  cos  —  +y  sin  —  =  e 


means  rotation,  in  positive  direction,  by  1  /  n  of  a  revolution, 


496  APPENDIX  I. 

multiplication  with  (—1)  means  reversal,  or  rotation  by  180°, 
multiplication  with  (+y )  means  positive  rotation  by  90°, 
multiplication  with  (— /)  means  negative  rotation  by  90°. 

307.    Complex  imaginary  plane  : 

While  the  positive  and  negative  numbers  can  be  rep- 
resented by  the  points  of  a  line,  the  complex  imaginary 
numbers  are  represented  by  the  points  of  a  plane,  with  the 
horizontal  axis  A'  O  A  as  real  axis,  the  vertical  axis  Br  O  B 
as  imaginary  axis.  Thus  all 

the  positive  real  numbers  are  represented  by  the  points  of  half 

axis  OA  towards  the  right ; 
the  negative  real  numbers  are  represented  by  the  points  of  half 

axis  OA'  towards  the  left ; 
the  positive  imaginary  numbers  are  represented  by  the  points  of 

half  axis  OB  upwards  ; 
the  negative  imaginary  numbers  are  represented  by  the  points  of 

half  axis  OB'  downwards  ; 
the  complex  imaginary  numbers  are  represented  by  the  points 

outside  of  the  coordinate  axes. 


APPENDIX    II. 


OSCILLATING    CURRENTS. 

INTRODUCTION. 

308.  An  electric  current  varying  periodically  between 
constant  maximum  and  minimum  values,  —  that  is,  in  equal 
time  intervals  repeating  the  same  values,  —  is  called  an 
alternating  current  if  the  arithmetic  mean  value  equals 
zero ;  and  is  called  a  pulsating  current  if  the  arithmetic 
mean  value  differs  from  zero. 

Assuming  the  wave  as  a  sine  curve,  or  replacing  it  by 
the  equivalent  sine  wave,  the  alternating  current  is  charac- 
terized by  the  period  or  the  time  of  one  complete  cyclic 
change,  and  the  amplitude  or  the  maximum  value  of  the 
current.  Period  and  amplitude  are  constant  in  the  alter- 
nating current. 

A  very  important  class  are  the  currents  of  constant 
period,  but  geometrically  varying  amplitude ;  that  is,  cur- 
rents in  which  the  amplitude  of  each  following  wave  bears 
to  that  of  the  preceding  wave  a  constant  ratio.  Such 
currents  consist  of  a  series  of  waves  of  constant  length, 
decreasing  in  amplitude,  that  is  in  strength,  in  constant 
proportion.  They  are  called  oscillating  currents  in  analogy 
with  mechanical  oscillations,  —  for  instance  of  the  pendu- 
lum,—  in  which  the  amplitude  of  the  vibration  decreases 
in  constant  proportion. 

Since  the  amplitude  of  the  oscillating  current  varies, 
constantly  decreasing,  the  oscillating  current  differs  from 

497 


498 


APPENDIX  II. 


the  alternating  current  in  so  far  that  it  starts  at  a  definite 
time,  and  gradually  dies  out,  reaching  zero  value  theoreti- 
cally at  infinite  time,  practically  in  a  very  short  time,  short 
even  in  comparison  with  the  time  of  one  alternating  half- 
wave.  Characteristic  constants  of  the  oscillating  current 
are  the  period  T  or  frequency  N  =  1/7",  the  first  ampli- 
tude and  the  ratio  of  any  two  successive  amplitudes,  the 
latter  being  called  the  decrement  of  the  wave.  The  oscil- 
lating current  will  thus  be  represented  by  the  product  of 


V 

^  ! 

I"**' 

\ 

^ 

-. 

\ 

/ 

S 

r~~ 

-- 

__ 

1 

>  \ 

180 

/ 

3W 

\ 

MO 

^ 

^-1 

raT 

X 

— 

— 

TWO 

—  J 

j»W8Q 

\ 

/ 

\ 

. 

___ 

^. 

•^-i 

\ 

/ 

_^ 

— 

->T=- 

Vy 

/.\ 

-' 

-~ 

t 

0 

en 

atin 
. 

g  E 
135 
cc 

M.F 

X 

"    [ 

E 

=5 

^ 
stf> 

1 

4afs 

2° 

a  periodic  function,  and  a  function  decreasing  in  geometric 
proportion  with  the  time.  The  latter  is  the  exponential 
function  Af~gt. 

309.   Thus,   the  general  expression   of  the  oscillating 
current  is 

/=  ^/-0'COS  (2-rrNt  —  S), 

since  A'-**  =  A' A-'*  =  U~bt. 

Where  e  =  basis  of  natural  logarithms,  the  current  may 
be  expressed 

7=  i(.~bt  cos  (2-n-JVf—  «)  =  ze-a*  cos  (<#>  -  £), 

where  <#>  =  %-nNt;  that  is,  the  period  is  represented  by  a 
complete  revolution. 


OSCILLATING    CURRENTS. 


499 


In  the  same  way  an  oscillating  electromotive  force  will 
be  represented  by 

E  =  etra*  cos  O  —  5). 

Such  an  oscillating  electromotive  force  for  the  values 
e  =  5,     a  =  .1435  or  «- 2™  =  .4,     £  =  0, 

is  represented  in  rectangular  coordinates  in  Fig.  207,  and 
in  polar  coordinates  in  Fig.  208.  As  seen  from  Fig.  207, 
the  oscillating  wave  in  rectangular  coordinates  is  tangent 
to  the  two  exponential  curves, 


Fig.  208. 


310.  In  polar  coordinates,  the  oscillating  wave  is  repre- 
sented in  Fig.  208  by  a  spiral  curve  passing  the  zero  point 
twice  per  period,  and  tangent  to  the  exponential  spiral, 


The  latter  is  called  the  envelope  of  a  system  O.L  oscillat- 
ing waves  of  which  one  is  shown  separately,  with  the  same 
constants  as  Figs.  207  and  208,  in  Fig.  209.  Its  character- 


500 


APPENDIX  II. 


istic  feature  is  :    The  angle  which    any  concentric  circle 
makes  with  the  curve  y  —  ee~a<t>,  is 


tan  a  = 


which  is,  therefore,  constant  ;  or,  in  other  words  :  "  The 
envelope  of  the  oscillating  current  is  the  exponential  spiral, 
which  is  characterized  by  a  constant  angle  of  intersection 


Fig.  209. 


Fig.  210. 


with  all  concentric  circles  or  all  radii  vectores."  The  oscil- 
lating current  wave  is  the  product  of  the  sine  wave  and  the 
exponential  or  loxodromic  spiral. 

311.    In    Fig.    210    let  j/  =  e€~a<t>   represent    the    expo-' 
nential  spiral  ; 

let  z  =  e  cos  (<£  —  a) 

represent  the  sine  wave  ; 
and  let  E  =  ef.-**  cos  (<£  —  w) 

represent  the  oscillating  wave. 

We  have  then 


tan  y3  = 


Ed* 
_  —  sin  (<£  —  w)  —  a  cos 

COS  (<£  —  oi) 
=  —  {tan  (<^>  —  £)  +  a} ; 


—  to) 


OSCILLATING   CURRENTS.  501 

that  is,  while  the  slope  of  the  sine  wave,  z  =  e  cos  (<£  —  w), 
is  represented  by 

tan  y  =  —  tan  (<£  —  w), 
the  slope  of  the  exponential  spiral  y  =  ei'0*  is 

tan  a  =  —  a  =  constant. 
That  of  the  oscillating  wave  E  =  *?e~a*  cos  (<£  —  to)  is 

tan  /3  =  —  {tan  (<£  —  w)  +  a}  . 

Hence,  it  is  increased  over  that  of  the  alternating  sine 
wave  by  the  constant  a.  The  ratio  of  the  amplitudes  of 
two  consequent  periods  is 


A  is  called  the  numerical  decrement  of  the  oscillating 
wave,  a  the  exponential  decrement  of  the  oscillating  wave, 
a  the  angular  decrement  of  the  oscillating  wave.  The 
oscillating  wave  can  be  represented  by  the  equation 

£  =  ec-**™"  cos  ($  —  5). 

In  the  instance  represented  by  Figs.  181  and  182>  we 
have  A  =  .4,  a  =  .1435,  a  =  8.2°. 


Impedance  and  Admittance. 

312.  In  complex  imaginary  quantities,  the  alternating 
wave  *  =  e  cos  (*  -  ffl) 

is  represented  by  the  symbol 

E  =  e  (cos  w  -\-j  sin  w)  =  <?x  -\-jez  . 

By  an  extension  of  the  meaning  of  this  symbolic  ex- 
pression, the  oscillating  wave  E  =  ee~a<t>  cos  (<f>  —  w)  can 
be  expressed  by  the  symbol 

E  =  e  (cos  w  -\-j  sin  w)  dec  a  =  (e±  -\-j'e^)  dec  a, 
where  a  =  tan  a  is  the  exponential  decrement,  a  the  angular 
decrement,  e~27ra  the  numerical  decrement. 


502  APPENDIX  II. 

Inductance. 

313.    Let    r  =  resistance,    L  =  inductance,    and  x  = 
2  IT  N  L  =  reactance. 

In  a  circuit  excited  by  the  oscillating  current, 

/=  /£-«*  cos  (<£  —  w)  =  /(cos  to  +y  sin  w)  dec  a  = 

(*i  -\-J*z)  dec  a, 
where  /i  =  /  cos  w,     /2  =  /  sin  £>,     a  =  tan  a. 

We  have  then, 

The  electromotive  force  consumed  by  the  resistance  r  of 
the  circuit  ^ 


The  electromotive  force  consumed  by  the  inductance  L 
of  the  circuit, 

Ef**L—~*iNI&t  =  *—. 

dt  d<$>         d<$> 

Hence  Ex  =  —  xif.~a^>  (sin  (<J>  —  fy  -\-  a  cos  (<£  —  w)} 

xi(.~a^    .     ,.        „    ,     N 

= sin  (^>  —  w  -f-  a). 

COS  a 

Thus,  in  symbolic  expression, 


£x  =  - °^—{—  sin  (w  —  a)  +/ cos  (w  —  a)}  dec  a 

COS  a 

=  —  x  i  (a  -f  y  )  (cos  w  +  7  sin  a>)  dec  a ; 
that  is,         Ex  =  —  x  I  (a  +/')  dec  a . 

Hence  the  apparent  reactance  of  the  oscillating  current 
circuit  is,  in  symbolic  expression, 

X  =  x  (a  +y')  dec  a. 

Hence  it  contains   an   energy  component  ax,  and   the 
impedance  is 

Z  =  (r  —  X)  dec  a  =  {r  —  x  (a  +/')}  dec  a  =  (r  —  ax  —jx)  dec  a. 

Capacity. 

314.   Let  r  =  resistance,  C  =  capacity,  and  xc  =  1  /2-n-JVC 
=  capacity  reactance.     In  a  circuit  excited  by  the  oscillating 


OSCILLATING   CURRENTS.  503 

current  /,  the  electromotive  force  consumed  by  the  capacity 
Cis 


or,  by  substitution, 

Ex  =  x  I  *  e~a*  cos  (<£ 

{sin  (<£  —  w)  —  a  COS  (<£  —  oi 


2 


(1  +  02)  COS  a 

hence,  in  symbolic  expression, 


sin  (</>  —  u>  —  a)  ; 


=  2  («  +  /)  (cos  w  +y  sin  w)  dec  a  ; 

hence, 


that   is,  the  apparent  capacity  reactance  of  the  oscillating 
circuit  is,  in  symbolic  expression, 


dec 


315.    We  have  then: 

In  an  oscillating  current  circuit  of  resistance  r,  induc- 
tive reactance  x,  and  capacity  reactance  xc  ,  with  an  expo- 
nential decrement  a,  the  apparent  impedance,  in  symbolic 
expression,  is  : 


*' 


1  +a2/         V         1  +** 

=  ra  —  jxa; 


504  APPENDIX  77. 

and,  absolute, 


Admittance. 
316.    Let    /=/e-a*cos^_£)==current< 

Then  from  the  preceding  discussion,  the  electromotive  force 
consumed  by  resistance  r,  inductive  reactance  x,  and  capa- 
city reactance  xc  ,  is 


cos  $  —         r  —  ax  —  a*e    —  sin  (<£  — 


=  iza(.~a^  cos  (<£  —  w  +  8), 
where     tan  8  = i_^ , 


a 
r  —  ax  —  —. -Xf 


substituting  &  +  8  for  G,  and  ^  =  /^a  we  have 


cos    <>  — 


I  =  —  e~a*  cos  (<#>  —  w  —  8) 


,1   \  cos  8         /  i        ~\    i    sin  8    .     /  , 
=  e e.    a<p  \  cos  (9  —  to )  -j sin  (9  — 


hence  in  complex  quantities, 

E  =  e  (cos  u>  -\-j  sin  oi)  dec  a, 
+    sin 


OSCILLATING   CURRENTS.  505 

or,  substituting, 


r  —  ax  — 
I  =E 


I-  dec  a. 


317.    Thus   in  complex  quantities,  for  oscillating  cur- 
rents, we  have  :  conductance, 


susceptance, 


admittance,  in  absolute  values, 

/ o i To  1 


in  symbolic  expression, 


Y=g+J» 


1  +  a2/        \  1  +  a2    ' 

Since  the  impedance  is 

Z  =  ir  —  ax  — 
we  have 


506  APPENDIX  II. 

that  is,  the  same  relations  as  in  the  complex  quantities  in 
alternating-current  circuits,  except  that  in  the  present  case 
all  the  constants  ra  ,  xa  ,  za  ,  g,  z,  y,  depend  upon  the  dec- 
rement a. 

Circuits  of  Zero  Impedance, 

318.  In  an  oscillating-current  circuit  of  decrement  a,  of 
resistance  r,  inductive  reactance  x,  and  capacity  reactance  xc, 
the  impedance  was  represented  in  symbolic  expression  by 


-jxa  = 


!  +  «» 

or  numerically  by 


Thus  the  inductive  reactance  x,  as  well  as  the  capacity 
reactance  xc,  do  not  represent  wattless  electromotive  forces 
as  in  an  alternating-current  circuit,  but  introduce  energy 
components  of  negative  sign 

a 

—  ax  —  -  -  x  : 
1  +  a2 

that  means, 

"  In  an  oscillating-current  circuit,  the  counter  electro- 
motive force  of  self-induction  is  not  in  quadrature  behind 
the  current,  but  lags  less  than  90°,  or  a  quarter  period;  and 
the  charging  current  of  a  condenser  is  less  than  90°,  or  a 
quarter  period,  ahead  of  the  impressed  electromotive  force." 

319.  In  consequence  of  the  existence  of  negative  en- 
ergy components  of  reactance  in  an  oscillating-current  cir- 
cuit, a  phenomenon  can  exist  which  has  no  analogy  in  an 
alternating-current  circuit  ;  that  is,  under  certain  conditions 
the  total  impedance  of  the  oscillating-current  circuit  can 
equal  zero  : 


In  this  case  we  have 


r  -  ax 


0  ;    x  --  ^—  =  0, 


-  —    c 
1  +  a2  1  +  fla 


OSCILLATING   CURRENTS.  507 


substituting  in  this  equation 

x  =  2  TT  NL  •  xc  = 
and  expanding,  we  have 
a 


That  is, 

"  If  in  an  oscillating-current  circuit,  the  decrement 
1 


and  the  frequency  N  =  r/4iraL,  the  total  impedance  of 
the  circuit  is  zero ;  that  is,  the  oscillating  current,  when 
started  once,  will  continue  without  external  energy  being 
impressed  upon  the  circuit." 

320.  The  physical  meaning  of  this  is  :  "  If  upon  an 
electric  circuit  a  certain  amount  of  energy  is  impressed 
and  then  the  circuit  left  to  itself,  the  current  in  the  circuit 
will  become  oscillating,  and  the  oscillations  assume  the  fre- 
quency N  =  r/4:7raL,  and  the  decrement 

1 


That  is,  the  oscillating  currents  are  the  phenomena  by 
which  an  electric  circuit  of  disturbed  equilibrium  returns  to 
equilibrium. 

This  feature  shows  the  origin  of  the  oscillating  currents, 
and  the  means  to  produce  such  currents  by  disturbing 
the  equilibrium  of  the  electric  circuit  ;  for  instance,  by 
the  discharge  of  a  condenser,  by  make  and  break  of  the 
circuit,  by  sudden  electrostatic  charge,  as  lightning,  etc. 
Obviously,  the  most  important  oscillating  currents  are 


508  APPENDIX  II. 

those  flowing  in  a  circuit  of  zero  impedance,  representing 
oscillating  discharges  of  the  circuit.  Lightning  strokes 
usually  belong  to  this  class. 

Oscillating  Discharges. 

321.    The    condition    of    an    oscillating    discharge    is 

Z  =  0,  that  is, 

~  ~        /  .1   r 

2aL       2Z~  ~1' 


If  r  =  0,  that  is,  in  a  circuit  without  resistance,  we  have 
a  =  0,  Af  =  1  /  2  TT  VZT  ;  that  is,  the  currents  are  alter- 
nating with  no  decrement,  and  the  frequency  is  that  of 
resonance. 

If  4  H  r2  C  -  1  <  0,  that  is,  r  >  2  V2T/T,  a  and  N 
become  imaginary  ;  that  is,  the  discharge  ceases  to  be  os- 
cillatory. An  electrical  discharge  assumes  an  oscillating 
nature  only,  if  r  <  2  V/,  /  C.  In  the  case  r  =  2  VZ,  /  C  we 
have  «  =  oo  ,  ./V  =  0  ;  that  is,  the  current  dies  out  without 
oscillation. 

From  the  foregoing  we  have  seen  that  oscillating  dis- 
charges, —  as  for  instance  the  phenomena  taking  place  if 
a  condenser  charged  to  a  given  potential  is  discharged 
through  a  given  circuit,  or  if  lightning  strikes  the  line 
circuit,  —  are  denned  by  the  equation  :  Z  =  0  dec  a. 

Since 

/    =  (/V+y/a)  dec  a,  Er  =  Ir  dec  a, 

Ex  =  -x  I  (a  +/)  dec  a,      Exc=  _^L_/(-  a  +/)  dec  a, 

we  have  r-aX--^—Xc  =  ^ 

I  +  a? 


hence,  by  substitution, 

Exc=  x  /(—  a  +/)  dec  a. 


OSCILLATING    CURRENTS.  50  £' 

The  two  constants,  t\  and  z'2,  of  the  discharge,  are  deter- 
mined by  the  initial  conditions,  that  is,  the  electromotive 
force  and  the  current  at  the  time  t  =  0. 

322.  Let  a  condenser  of  capacity  C  be  discharged 
through  a  circuit  of  resistance  r  and  inductance  L.  Let 
e  =  electromotive  force  at  the  condenser  in  the  moment 
of  closing  the  circuit,  that  is,  at  the  time  t  —  0  or  <£  =  0. 
A.t  this  moment  the  current  is  zero  ;  that  is, 

7=//2,     /1==0. 
Since          Exe=  •*/(—  a  +/)  dec  a  =  e  at  <f>  =  0, 

we  have     x  /2  Vl  +  a2  =  e  or  /2  = = . 

x  V 1  +  a2 
Substituting  this,  we  have, 

I  —j  —    e          dec  a,  Er  =je r          dec  a, 

x  Vl  +  a2  x  Vl  +  az 

Ex  =         e        (1  -ja)  dec  a,   ^c= e         (1  +/ «)  dec  a, 

Vl  +  «8  Vl  +  a2 

the  equations  of  the  oscillating  discharge  of  a  condense 
of  initial  voltage  e. 

Since  x  =  2  *•  N  L, 

1 


we  have 

x  = 


hence,  by  substitution, 

l 

—  dec  a, 


.510  APPENDIX  II. 

E   -        ef\fC 

-f^r-,  —  —  rr~  \/  ~r~ 


47TZ 


the  final  equations  of  the  oscillating  discharge,  in  symbolic 
expression. 

Oscillating  Current   Transformer. 

323.  As  an  instance  of  the  application  of  the  symbolic 
method  of  analyzing  the  phenomena  caused  by  oscillating 
currents,  the  transformation  of  such  currents  may  be  inves- 
tigated. If  an  oscillating  current  is  produced  in  a  circuit 
including  the  primary  of  a  transformer,  oscillating  currents 
will  also  flow  in  the  secondary  of  this  transformer.  In  a 
transformer  let  the  ratio  of  secondary  to  primary  turns  be/. 
Let  the  secondary  be  closed  by  a  circuit  of  total  resistance, 
i\=  r{  -\-  TJ",  where  1\  =  external,  1\'  =  internal,  resistance. 
The  total  inductance  Ll  =  Z/  -f  /,/',  where  Z/  =  external, 
Zj"  =  internal,  inductance  ;  total  capacity,  Cv  Then  the 
total  admittance  of  the  secondary  circuit  is 

)  dec  a  = 


where  xl=  2irJVLl=  inductive  reactance:  xcl  =  \l1-jrNC  '  = 
capacity  reactance.  Let  rQ  =  effecive  hysteretic  resistance, 
Z  =  inductance  ;  hence,  x^  =  Z-n-N  LQ  =  reactance  ;  hence, 


admittance 


of  the  primary  exciting  circuit  of  the  transformer  ;  that  is, 
the  admittance  of  the  primary  circuit  at  open  secondary 
circuit. 

As  discussed  elsewhere,  a  transformer  can  be  considered 
as  consisting  of  the  secondary  circuit  supplied  by  the  im- 
pressed electromotive  force  over  leads,  whose  impedance  is 


OSCILLATING   CURRENTS.  511 

equal  to  the  sum  of  primary  and  secondary  transformer  im- 
pedance, and  which  are  shunted  by  the  exciting  circuit,  out- 
side of  the  secondary,  but  inside  of  the  primary  impedance. 
Let  r  =  resistance  ;  L  =  inductance  ;    C  =  capacity  ; 

hence'  x  =  2  TT  NL  =  inductive  reactance, 

xc  =  1  /  2  TT  N  C  =  capacity  reactance  of  the  total  primary 
circuit,  including  the  primary  coil  of  the  transformer.  If 
EI  =  EI  dec  a  denotes  the  electromotive  force  induced  in 
the  secondary  of  the  transformer  by  the  mutual  magnetic 
flux ;  that  is,  by  the  oscillating  magnetism  interlinked 
with  the  primary  and  secondary  coil,  we  have  Iv  =  E^  Yl 
dec  a  =  secondary  current. 

Hence,  //  =  /  7X  dec  a  =  pEJ  Yl  dec  a  =  primary  load 
current,  or  component  of  primary  current  corresponding  to 

secondary  current.     Also,  70  =  -  2j/  F0   dec  a  =  primary 

/  ' 
exciting  current ;  hence,  the  total  primary  current  is 

/=  //  +  70  =  £-'{Fo  +/2  Y,}  dec  a. 

E' 
E'  =  -^-i-  dec  a  =  induced  primary  electromotive  force. 

/ 
Hence  the  total  primary  electromotive  force  is 

E  =  (£'  +  /Z)  dec  a  =  £L  (1  +  Z  F0  +/2Z  Y,}  dec  a. 
P 

In    an   oscillating  discharge   the   total   primary  electro- 
motive force  E  =  0  ;  that  is, 


or,  the  substitution 

a 


1  + 


(r0  -  ax0)  -. 


.  0. 


512  APPENDIX  II. 


Substituting  in  this  equation,  ^r=2  it  N  C,  xc  =  ~L/'2 
etc.,  we  get  a  complex  imaginary  equation  with  the  two 
constants  a  and  N.  Separating  this  equation  in  the  real 
and  the  imaginary  parts,  we  derive  two  equations,  from 
which  the  two  constants  a  and  N  of  the  discharge  are 
calculated. 

324.  If  the  exciting  current  of  the  transformer  is  neg- 
ligible, —  that  is,  if  YQ  =  0,  the  equation  becomes  essentially 
simplified,  — 


I  a          \         .  I  x      \ 

(r  —  a  x xc  1  —  j  (  x —  I 

1+/2v 1  +  *8   i v Ljt^l=0; 


that  is, 


or,  combined,  — 

(r,  -2aXl)  +/2  (r-2  ax)  =  0, 


Substituting  for  xlt  x,  xel,  xei  we  have 


+/aZ) 


i+/V        /        4(A+/ 
+/2Z)  V  (n  +/V)2  (Ci 

!}  dec  a, 


7  =pEi   YI  dec  a, 
/!  =  ^/  F!  dec  a, 

the  equations  of  the  oscillating-current  transformer,  with 
E{  as  parameter. 


INDEX. 


PAGE 

Addition 494.  498 

Admittance,  conductance,  suscep- 

tance,  Chap.  vn.       ...     52 

definition 53 

parallel  connection       ...     57 
primary    exciting,    of    trans- 
former       204 

of  induction  motor  .  .  .  240 
Advance  of  phase,  hysteretic  .  .115 
Algebra  of  complex  imaginary 

Quantities,  App.  I.  .  .  .  489 
Alternating  current  generator, 

Chap,  xvii 297 

transformer,  xiv 193 

motor,    commutator,    Chap. 

xx 354 

motor,    synchronous,    Chap. 

xix 321 

Alternating  wave,  definition     .     .     11 
general       ..."...     .       7 
Alternators,  Chap.  xvii.     .     .     .  297 
parallel      operation,      Chap. 

xvin 311 

series  operation 313 

synchronizing,  Chap.  xvin.  .  311 
synchronizing  power  in  paral- 
lel operation 317 

Ambiguity  of  vectors  ....  43 
Amplitude  of  alternating  wave  .  7 
Angle  of  brush  displacement  in 

repulsion  motor  ....  361 
Apparent  total  impedance  of 

transformer 208 

Arc,  distortion  of  wave  shape  by  394 

power  factor  of 395 

Arithmetic  mean  value,  or  average 

value  of  alternating  wave  11 
Armature  reaction  of  alternators 

and  synchronous  motors  .  297 
51 


Armature  reaction  of  alternators, 
as  affecting  parallel  opera- 
tion   313 

self-induction  of   alternators 

and  synchronous  motors   .  300 
slots,   number    of,   affecting 
wave  shape 384 

Associate  numbers 494 

Asynchronous,  see  induction  .     . 

Average  value,  or  mean  value  of 

alternating  wave  ....     11 

Balance,  complete,  of  lagging 
currents  by  shunted  con- 

densance 74 

Balanced  and  unbalanced  poly- 
phase systems,  Chap. 

xxvii 440 

Balanced  polyphase  system     .     .  431 
quarter-phase  system  .     .     .  484 
three-phase  system       .     .     .  481 
Balance  factor  of  polyphase  sys- 
tem     441 

of  lagging  currents  by  shun- 
ted condensance       ...     75 
Biphase,  see  quarter-phase       .     . 

Cables,  as  distributed  capacity    .  158 
with  resistance  and  capacity 
topographic  circuit  charac- 
teristic       47 

Calculation   of    magnetic   circuit 

containing  iron     .     .     .     •  125 
of  constant  frequency  induc- 
tion generator      ....  269 
of  frequency  converter     .     .  232 
of  induction  motor       .     .     .  262 
of  single-phase  induction  mo- 
tor     .  .  287 


514 


INDEX. 


Calculation  of  transmission  lines, 

Chap,  ix 83 

Capacity    and    inductance,     dis- 
tributed, Chap.  xin.     .     .  158 
as  source  of  reactance      .     .       6 
in   shunt,  compensating   for 

lagging  currents  ....     72 
intensifying  higher   harmon- 
ics       402 

see  condenser   and    conden- 

sance. 
Chain   connection    of    induction 

motors,  or  concatenation  .  274 
Characteristic    circuit    of    cable 
with  resistance  and  capa- 
city      48 

circuit    of  transmission  line 
with  resistance,  inductance, 
capacity,  and  leakage    .     .     49 
curves  of  transmission  lines  .  172 
field  of  alternator   ....  304 
power  of  polyphase  systems  447 
Circuit    characteristic    of    cable 
with  resistance  and  capa- 
city     48 

characteristic  of  transmission 
line  with  resistance,  induc- 
tance, capacity  and  leakage    49 
factor  of  distorted  wave  .     .  415 
with  series  impedance      .     .     68 
with  series  reactance    ...     61 
with  series  resistance  ...     58 
Circuits  containing  resistance,  in- 
ductance,    and     capacity, 

Chap,  vin 58 

Coefficient  of  hysteresis  .     .  116 
Combination  of  alternating  sine 
waves  by  parallelogram  or 
polygon  of  vectors    ...     21 
of  double  frequency  vectors, 

as  power 163 

of  sine  waves  by  rectangular 

components 35 

of   sine   waves   in    symbolic 

representation      ....     38 
Commutator  motor,  Chap.  xx.  354 


Compensation    for    lagging   cur- 
rents by  shunted  conden- 

sance 72 

Complete   diagram   of   transmis- 
sion line  in  space      .     .     .192 
Complex  imaginary  number    .     .  492 
imaginary  quantities,  algebra 

of,  App.  i 489 

imaginary  quantities,  as  sym- 
bolic representation  of  al- 
ternating waves  ....  37 

quantity  Chap,  v 33 

Compounding  curve  of  frequency 

converter 232 

Concatenated  couple  of  induction 

motors,  calculation  .     .     .  276 
Concatenation  of   induction  mo- 
tors     274 

Condensance  in  shunt,  compen- 
sating for  lagging  currents     72 
in  symbolic  representation    .     40 
or  capacity  reactance  ...       6 
see  capacity  and  condenser 
Condensers,   distortion  of  wave 

shape  by 393 

see  capacity  and  condensance 
with  distorted  wave     .     .     .  419 
with   single-phase   induction 

motor 286 

Conductance,  effective,  definition  104 
in    alternating    current    cir- 
cuits, definition    ....     54 
in    continuous    current    cir- 
cuits   52 

of  receiver  circuit,  affecting 

output  of  inductive  line  .  89 
parallel  connection  ...  52 
see  resistance 

Conjugate  numbers 494 

Constant  current  —  constant  po- 
tential transformation  .     .     76 
current,     constant    potential 
transformation    by    trans- 
mission line 181 

potential,  constant  current 
transformation  .  .  76 


INDEX. 


515 


Constant  potential,  constant  cur- 
rent transformation  by 
transmission  line ....  181 

rotating  M  M.F 436 

Constants,   characteristic,   of   in- 
duction motor      ....  262 
Continuous  current  system,  distri- 
bution efficiency  ....  473 
Control,  by  change  of  phase,  of 

transmission  line,  Chap.  ix.     83 
of  receiver  circuit  by  shunted 

susceptance 96 

Converter    of    frequency,   Chap. 

xv 219 

Counter  E.M.F.  constant  in  syn- 
chronous motor  ....  349 

of  impedance 25 

of  inductance 25 

of  resistance .25 

of  self-induction      .....     24 
Counting  or  numeration      .     .     .  489 
Cross-flux,    magnetic,    of    trans- 
former     193 

of  transformer,  use  for  con- 
stant   power  or   constant 
current  regulation     .     .     .  194 
Current,   minimum,   in    synchro- 
nous motor 345 

waves,  alternating,  distorted 

by  hysteresis 109 

Cycle,  or  complete  period   ...     10 

Decrement  of  oscillating  wave     .  501 
Delta  connection  of  three-phase 

system 453 

current  in  three  phase  system  455 
potential  of  three-phase  sys- 
tem      455 

Y  connection  of  three-phase 

transformation      ....  463 
Demagnetizing  effect  of  armature 
reaction  of  alternators  and 
synchronous  motors      .     .  298 
effect  of  eddy  currents      .     .  136 
Dielectric  and  electrostatic   phe- 
nomena .  .  144 


Dielectic  and  electrostatic  hyste- 
resis   145 

Diphase,  see  quarter-phase. 
Discharge,  oscillating     ....  508 
Displacement  angle  of  repulsion 

motor 361 

of  phase,  maximum,  in  syn- 
chronous motor   ....  347 
Distorted  wave,  circuit  factor      .  415 
wave,   decreasing  hysteresis 

loss 407 

wave,   increasing    hysteresis 

loss 407 

wave  of  condenser  ....  419 
wave  of  synchronous  motor .  422 
wave,  some  different  shapes  .  401 
wave,   symbolic    representa- 
tion, Chap.  xxiv.      .     .     .  410 
wave,  in  induction  motor  .     .  426 
Distortion  of  alternating  wave     .       9 
of  wave  shape  and  eddy  cur- 
rents   408 

of  wave  shape,  and  insulation 

strength 409 

of  wave  shape  and  its  causes, 

Chap,  xxn 383 

of  wave  shape  by  hysteresis  .  109 
of    wave   shape,   effect    of, 

Chap,  xxin 398 

of  wave  shape,  increasing  ef- 
fective value 405 

Distributed  capacity,  inductance, 
resistance,  and  leakage, 

Chap,  xni 158 

Distribution  efficiency  of  systems.  468 
Divided    circuit,    equivalent     to 

transformer 209 

Division 491,494 

Double  delta  connection  of  three- 
phase —  six-phase  transfor- 
mation   465 

frequency  quantities,  as  pow- 
er, Chap,  xii 150 

frequency  values  of  distorted 
wave,  symbolic  representa- 
tion .  .  413 


516 


INDEX. 


Double  peaked  wave 399 

saw-tooth  wave 399 

T  connection  of  three-phase 

—  six-phase      transforma- 
tion     466 

Y  connection  of  three-phase 

—  six-phase      transforma- 
tion     466 

.Eddy     currents,     unaffected    by 

wave-shape  distortion   .     .  408 
demagnetizing   or   screening 

effect 136 

in   conductor,   and    unequal 

current  distribution  .     .     .  139 
Eddy  or  Foucault  currents,  Chap. 

xi 129 

Effective   reactance  and  suscep- 

tance,  definition  ....  105 
resistance  and  conductance, 

definition 104 

resistance      and     reactance, 

Chap,  x 104 

to  maximum  value  ....  14 
value  of  alternating  wave  .  11 
value  of  alternating  wave, 

definition 14 

value  of  general  alternating 

wave 15 

Effects     of     higher     harmonics, 

Chap,  xxin 398 

Efficiency,   maximum,   of  induc- 
tive line 93 

Efficiency  of  systems,  Chap.  xxx.  468 
Electro-magnetic    induction,  law 

of,  Chap.  Ill 16 

.Electrostatic  and  dielectric  phe- 
nomena   144 

hysteresis 145 

Energy  component  of  self-induc- 
tion     372 

flow  of,  in  polyphase  system,  441 
Epoch  of  alternating  wave  ...       7 
Equations,  fundamental,  of  alter- 
nating current  transformer, 

208,  225 


Eauations,  fundamental,  of  gen- 
eral alternating  current 
transformer,  or  frequency 

converter 224 

of  induction  motor  .     .    226,  242 
of  synchronous  motor  .     .     .  339 
of  transmission  line      .     .     .  169 
Equations,  general,  of  apparatus, 
see  equations,fundamental. 
Equivalence  of  transformer  with 

divided  circuit 209 

Equivalent  sine  wave  of  distorted 

wave in 

Evolution 491,  495 

Exciting  admittance  of  induction 

motor 240 

admittance  of  transformer     .  204 
current  of   magnetic  circuit, 

distorted  by  hysteresis.     .111 
current  of  transformer      .     .  195 

Field  characteristic  of  alternator  .  304 
First  harmonic,  or  fundamental, 

of  general  alternating  wave,       8 
Five-wire  single-phase  system,  dis- 
tribution efficiency    .     .     .  470 

Flat-top  wave 399 

Flow  of  power  in  polyphase  sys- 
tem   441 

Foucault  or  Eddy  currents,  Ch.  xi.  129 
Four-phase,  see  quarter-phase. 

Fraction 491 

Free  oscillations  of  circuit  .     .     .  508 
Frequency  converter,  Chap.  xv.  .  219 
converter,  calculation  .     .     .232 
converter,  fundamental  equa- 
tions   224 

of  alternating  wave      ...       7 
ratio  of   general    alternating 
current  transformer  or  fre- 
quency converter      .     .     .  221 
Friction,  molecular  magnetic  .     .  106 
Fundamental  equations,  see  equa- 
tions, fundamental, 
frequency     of    transmission 
line  discharge      .     .     .     .186 


INDEX. 


517 


Fundamental  equations,  or  first 
harmonic  of  general  alter- 
nating wave 8 

General  alternating  current  trans- 
former, or  frequency  con- 
verter, Chap.  xv.  ...  219 

alternating  wave      .     .     .     .  7,  8 

alternating    wave,    symbolic 
representation,Chap.xxiv.  410 

equations,  see  equations,  fun- 
damental. 

polyphase     systems,     Chap. 

xxv 430 

Generator  action  of  concatenated 

couple 280 

of  reaction  machine      .     .     .377 

alternating     current,     Chap. 
xvn 297 

synchronous,  operating  with- 
out field  excitation   .     .     .  371 

induction 265 

induction,  calculation  for  con- 
stant frequency    ....  269 

reaction,  Chap.  xxi.     .     .     .371 

vector  diagram 28 

Graphical  construction  of  circuit 

characteristic    .     .     .     .  48,  49 
Graphic  representation,  Chap.  iv.     19 

limits  of  method      ....     33 

see  polar  diagram. 

Harmonics,    higher,    effects    of, 

Chap,  xxin 398 

higher,     resonance     rise     in 

transmission  lines     .     .     .  402 
of  general  alternating  wave  .       8 
Hedgehog  transformer   ....  195 
Hemisymmetrical  polyphase  sys- 
tem      439 

Henry,  definition  of 18 

Hexaphase,  see  six-phase. 

Hysteresis,  Chap,  x 104 

advance  of  phase     .     .     .     .115 
as  energy  component  of  self- 
induction     372 


Hysteresis,  coefficient     .     .     .     .116 

cycle  or  loop 107 

dielectric,  or  electrostatic  .  145 
energy  current  of  transformer  196 
loss,  effected  by  wave  shape,  407 
loss  in  alternating  field  .  .114 

magnetic 106 

motor 293 

of  magnetic  circuit,  calcula- 
tion     125 

or  magnetic  energy  current  .  115 

Imaginary  number 492 

quantities,  complex,  algebra 

of,  App.  1 489 

Impedance 2 

in  series  with  circuit  ...  68 
in  symbolic  representation  .  39 
primary  and  secondary,  of 

transformer 205 

see,  admittance. 

series  connection     ....     57 

total  apparent,  of  transformer  208 

Independent  polyphase  system    .  431 

Inductance 4 

definition  of 18 

factors  of  distorted  wave .     .  415 

mutual '  ...  142 

Induction,  electro-magnetic,  law  of     16 

electrostatic 147 

generator 265 

generator,     calculation     for 

constant    frequency       .     .  269 
generator,  driving  synchron- 
ous motor 272 

motor,  Chap,  xvi 237 

motor 281 

motor,  calculation    ....  262 
motor,  concatenation  or  tan- 
dem control 274 

motor,     fundamental     equa- 
tions      226,  242 

motor,    graphic    representa- 
tion  244 

motors  in  concatenation,  cal- 
culation .  ....  276 


518 


INDEX. 


Induction  motor,  synchronous    .  291 
motor  torque,  as  double  fre- 
quency vector      .     .     .     .156 
motor  with  distorted  wave   .  426 
Inductive  devices  for  starting  sin- 
gle-phase induction  motor    283 
line,  effect  of  conductance  of 
receiver   circuit   on   trans- 
mitted power 89 

line,  effect  of  susceptance  of 
receiver  circuit  on  trans- 
mitted power 88 

line,  in  symbolic  representa- 
tion   41 

line,  maximum    efficiency  of 

transmitted  power    ...     93 
line,    maximum    power  sup- 
plied over 87 

line,  maximum  rise  of  poten- 
tial by  shunted  suseeptance  101 
line,  phase  control  by  shunted 

susceptance 96 

line,  supplying  non-inductive 

receiver  circuit     ....     84 
Influence,  electrostatic    .     .     .     .147 
Instantaneous    values  and    inte- 
gral values,  Chap.  n.    .     .     11 
value  of  alternating  wave     .     1 1 
Insulation  strength  with  distorted 

wave 409 

Integral    values    of     alternating 

wave 11 

Intensity  of  sine  wave    ....     20 
Interlinked    polyphase    systems, 

Chap,  xxvin 452 

polyphase  system    ....  431 
Internal     impedance     of     trans- 
former     205 

Introduction,  Chap.  1 1 

Inverted  three-phase  system   .     .  434 
three-phase   system,  balance 

factor 443,  446 

three-phase  system,  distribu- 
tion efficiency 472 

Involution 490,495 

Iron,  laminated,  eddy  currents     .  131 


Iron  wire,  eddy  currents    .     .     .     133 
wire,  unequal  current  distri- 
bution  in   alternating  cir- 
cuit      142 

Irrational  number 492 

f,  as  imaginary  unit      ....  37 
introduction    of,    as    distin- 
guishing index      ....  36 
Joules's  law  of    alternating  cur- 
rents    6 

law  of  continuous  currents  .  1 

Kirchhoff's  laws    in   symbolic 

representation  ....  40 
laws  of  alternating  current 

circuits 58 

laws  of  alternating  sine  waves 

in  graphic  representation  .  22 
laws  of  continuous  current 

circuits 1 

Lagging  currents,  compensation 
for,  by  shunted  conden- 

sance 72 

Lag  of  alternating  wave     ...     21 
of  alternator  current,  effect 
on  armature  reaction  and 

self-induction 298 

Laminated  iron,  eddy  currents     .  131 
Law    of   electro-magnetic   induc- 
tion, Chap,  in 16 

L  connection  of  three-phase,  quar- 
ter-phase transformation    .  465 
connection     of     three-phase 

transformation      ....  464 
Lead  of  alternating  wave    ...     21 
of   alternator   current,  effect 
on  armature  reaction  and 
self-induction  .     .          .     .  298 
Leakage    current,    see    Exciting 

current. 

of  electric  current    ....   148 
Lightning  discharges  from  trans- 
mission lines,  frequencies 

181,  188 


INDEX. 


519 


Line,  inductive,  vector  diagram  .     23 
with  distributed  capacity  and 

inductance 158 

with  resistance,  inductance, 
capacity,  and  leakage, 
topographic  circuit  charac- 
teristic   49 

Logarithmation 491 

Long-distance  lines,  as  distributed 

capacity,  and  inductance     158 

Loxodromic  spiral 500 

Magnetic  circuit  containing  iron, 

calculation 125 

hysteresis 106 

or  hysteretic  energy  current .  116 

Magnetizing  current 115 

current  of  transformer     .     .  196 
effect    of   armature  reaction 
in  alternators  and  synchro- 
nous motors 298 

Main   and   teazer   connection  of 

three-phase  transformation  464 
Maximum  output  of  synchronous 

motor 342 

power  of  induction  motor      .  252 
power  of  synchronous  motor  342 
power  supplied   over  induc- 
tive line 87 

rise  of  potential  in  inductive 
line,  by  shunted  suscep- 

tance 101 

to  effective  value     ....     14 

to  mean  value 13 

torque  of  induction  motor    .  250 

value  of  alternating  wave     .     11 

Mean  to  maximum  value    ...     13 

value 12 

value,  or   average   value   of 

alternating  wave  .     .     .     .     11 
Mechanical   power   of  frequency 

converter 227 

Minimum  current  in  synchronous 

motor 345 

M.  M.  F.  of    armature     reaction 

of  alternator    .  .  297 


M.  M.  F.    rotating,   of   constant 

intensity 436 

's  acting  upon  alternator  ar- 
mature     297 

Molecular  magnetic  friction     .     .  106 
Monocyclic  connection  of  three- 
phase-inverted  three-phase 
transformation      ....  464 
devices    for   starting   single- 
phase  induction  motors     .  283 

systems 447 

Monophase,  see  Single-phase. 
Motor,   action   of     reaction    ma- 
chine   377 

alternating  series  ....  363 
alternating  shunt  ....  368 
commutator,  Chap.  xx.  .  .  354 

hysteresis 293 

induction,  Chap.  xvi.  .  .  .  237 
reaction,  Chap.  xxi.  .  .  .371 

repulsion 354 

single-phase  induction      .     .  281 
synchronous,  Chap,  xix   .     .  321 
synchronous,    driven   by   in- 
duction generator     .     .     .  272 
synchronous  induction     .     .  291 
Multiple  frequency   of   transmis- 
sion line  discharge    .     .     .  185 

Multiplication 490,494 

Mutual  inductance 142 

inductance     of    transformer 
circuits 194 

Natural  period  of  transmission 

line 181 

Negative  number 490 

Nominal  induced  E.M.F.  of  alter- 
nator   302 

Non-inductive     load     on     trans- 
former     212 

receiver  circuit  supplied  over 
inductive  line       ....     84 

N-phase    system,    balance    fac- 
tor       443 

phase  system,  symmetrical    .  435 

Numeration  or  counting      .     .     .  489 


520 


INDEX. 


Ohms  law  in  symbolic  represen- 
tation      40 

of  alternating  currents       .     .       2 

of  continuous  currents     .     .       1 

Oscillating  currents,  App.  n.  .     .  497 

discharge 508 

Oscillation  frequency  of  transmis- 
sion line 181 

Output,  see  Power. 

Overtones,  or   higher  harmonics 

of  general  alternating  wave      8 

Parallel  connection  of  conduc- 
tances     52 

Parallelogram  law  of  alternating 

sine  waves 21 

of  double-frequency  vectors, 

as  power 153 

Parallel  operation  of  alternators, 

Chap,  xviir 311 

Peaked  wave 399 

Period,   natural,  of  transm.   line  181 
of  alternating  wave      ...       7 
Phase  angle  of  transmission  line     171 
control,  maximum  rise  of  po- 
tential by 101 

control  of  inductive  line  by 

shunted  susceptance     .     .     96 
control  of  transmission  line, 

Chap,  ix 83 

difference  of 7 

displacement,    maximum,   in 

synchronous  motor  .     .     .  347 
of  alternating  wave      ...       7 

of  sine  wave 20 

splitting  devices  for  starting 
single-phase  induction  mo- 
tors   283 

Plane,  complex  imaginary  .     .     .  496 
Polar   coordinate   of   alternating 

waves 19 

diagram  of  induction  motor  244 
diagram  of  transformer     .     .  196 
diagram  of  transmission  line  191 
diagrams,  see  Graphic  repre- 
sentation. 


PAGE 

Polarization  as  capacity      .     .  6 

distortion  of  wave  shape  by  .  393 

Polycyclic  systems 447 

Polygon  of  alternating  sine  waves     22 
Polyphase  system,  balanced     .     .  431 
systems,  balanced  and  unbal- 
anced, Chap.  xxvn.      .     .  440 
.     systems,  efficiency   of   trans- 
mission, Chap.  xxx.     .     .  468 
systems,  flow  of  power    .     .  441 
systems,  general,  Chap.  xxv.  430 
systems,  hemisymmetrical     .  439 
systems,    interlinked,    Chap. 

xxvin 452 

systems,  symmetrical,  Chap. 

xxvi 435 

systems,  symmetrical    .     .     .  430 
systems,    symmetrical,    pro- 
ducing constant  revolving 

M.M.F 436 

systems,   transformation   of, 

Chap,  xxix 460 

systems,  unbalanced    .     .     .  431 
systems,  unsymmetrical     .     .  430 
Power    and      double     frequency 
quantities  in  general,  Chap. 

XII 150 

characteristic    of    polyphase 

systems 447 

characteristic  of  synchronous 

motor 341 

equation  of  alternating  cur- 
rents   6 

equation  of  alternating  sine 
waves  in  graphic  represen- 
tation   23 

equation  of  continuous  cur- 
rents   1 

factor  of  arc 395 

factor  of  distorted  wave    .     .  414 
factor  of  reaction  machine    .  381 
flow  of,  in  polyphase  system    441 
flow  of,  in  transmission  line    177 
maximum,  of  inductive   line 
with  non-inductive  receiver 
circuit     .  .     86 


INDEX. 


521 


PAGE 

Power,  maximum  of  synchronous 

motor 432 

maximum  supplied  over  in- 
ductive line 87 

of  complex  harmonic  wave  .  405 
of  distorted  wave  ....  413 
of  frequency  converter  .  .  227 
of  general  polyphase  system  459 
of  induction  motor  ....  246 
of  repulsion  motor  ....  360 
parallelogram  of,  in  symbolic 

representation       ....  153 
real  and  wattless,  in  symbol- 
ic representation       .     .     .  151 
Primary   exciting    admittance    of 

induction  motor  ....  240 
exciting  admittance  of  trans- 
former      204 

impedance  of  transformer     .  205 
Pulsating  wave,  definition  ...     11 
Pulsation  of  magnetic  field  caus- 
ing  higher   harmonics    of 

E.M.F 384 

of  reactance  of  alternator  ar- 
mature causing  higher  har- 
monics    .  391 

of  resistance,  causing  higher 
harmonics 393 

Quadriphase,  see  Quarter-phase. 
Quarter-phase,  five-wire    system, 

distribution  efficiency   .     .  471 
system,  Chap.  xxxn.    .     .     .  483 

system 43^ 

system,  balance  factor  .    442,  445 
system,  distribution  efficiency  471 
system,  symmetry    ....  436 
system,      transmission      effi- 
ciency      474 

three-phase  transformation  .  465 

unitooth  wave 388 

Quintuple  harmonic,  distortion  of 

wave  by 400 

Ratio  of  frequencies  in  general 
alternating  current  trans- 
former .  .  221 


Ratio  of  frequencies  of  transfor- 
mation of  transformer  .     .  207 

Reactance     2 

definition 18 

effective,  definition  .  .  .  105 
in  series  with  circuit  .  .  .  61 
in  symbolic  representation  .  39 
periodically  varying  .  .  .  373 
pulsation  in  alternator  caus- 
ing higher  harmonies  .  .  391 

sources  of 8 

synchronous,  of  alternator    .  301 
see  Susceptance. . 

Reaction  machines,  Chap.  xxi.    .  371 
machine,  power-factor       .     .  381 
armature,  of  alternator     .     .  297 
Rectangular  coordinates  of  alter- 
nating vectors      ....     34 
diagram    of     transmission 

line 191 

Reflected   wave  of   transmission 

line 169 

Reflexion  angle   of   transmission 

line 169 

Regulation   curve    of    frequency 

converter 232 

of    alternator    for    constant 

current .  309 

of    alternator    for    constant 

power 310 

of    alternator    for    constant 

terminal  voltage      .     .     .  308 
Reluctance,  periodically  varying  .  373 
pulsation  of,  causing  higher 
harmonics  of  E.M.F.    .     .  384 

Repulsion  motor 354 

motor,  displacement  angle  .     .     .  361 

motor,  power 360 

motor,  starting  torque       .     .  361 

motor,  torque 360 

Resistance     and       reactance    of 
transmission  Lines, 

Chap,  ix 83 

effective,  definition       .     .     .  104 
effective,  of  alternating  cur- 
rent circuit .  2 


522 


INDEX. 


22 


153 


Resistance  and  reactance  in  alter- 
nating current  circuits ...       2 
in  series  with  circuit     ...     58 
of  induction    motor    secon- 
dary,     affecting     starting 

torque 254 

pulsation,  causing  higher  har- 
monics     393 

series  connection     ....     52 
see  Conductance. 
Resonance  rise  by  series  induc- 
tance,  with    leading    cur- 
rent     65 

rise  in  transmission  lines  with 

higher  harmonics      .     .     .  402 
Resolution    of    alternating    sine 
waves    by    the    parallelo- 
gram or  polygon  of  vec- 
tors      

of  double  frequency  vectors, 

as  power 

of  sine  waves  by  rectangular 

components 35 

of  sine   waves   in   symbolic 

representation      ....     38 
Reversal  of  alternating  vector  by 

multiplication  with  —  1     .     36 
Revolving  magnetic  field    .     .     .  436 
M.  M.  F.  of  constant  inten- 
sity      436 

Ring   connection   of    interlinked 

polyphase  system     .     .     .  453 
current  of  interlinked   poly- 
phase system 455 

potential  of  interlinked  poly- 
phase system 455 

Rise   of  voltage   by  inductance, 

with  leading  current      .     .     62 
of  voltage  by  inductance  in 
synchronous  motor  circuit     65 

Roots  of  the  unit 495 

Rotating  magnetic  field  ....  436 
M.M.F.  of  constant  intensity  436 

Rotation 495 

by    90°,    by     multiplication 
with  ±  j 37 


Saturation,  magnetic,  effect  on 
exciting  current  wave   .     . 

Sawtooth  wave 

Screening  effect  of  eddy  currents 
Screw    diagram    of   transmission 

line 

Secondary   impedance   of   trans- 
former  . 


Self-excitation  of  alternator  and 
synchronous  motor  by  ar- 
mature reaction  .... 

Self-inductance 

E.M.F.  of 

of  transformer 

of   transformer  for  constant 
power  or  constant  current 

regulation 

Self-induction,  energy  component 

of 

of  alternator  armature      .     . 

reducing  higher  harmonics   . 

Series  connection  of  impedances 

of  resistances 

impedance  in  circuit     .     .     . 
motor,  alternating    .... 
operation  of  alternators   .     . 
reactance  in  circuit       .     .     . 
resistance  in  circuit 
Shunt  motor,  alternating 
Sine  wave 


circle  as  polar  characteristic 

equivalent,  of  distorted  wave, 
definition 

representation  by  complex 

quantity 

Single-phase  induction  motor 

induction  motor,  calculation 

induction  motor,  starting  de- 
vices   

induction  motor,  with  con- 
denser in  tertiary  circuit  . 

system,  balance  factor      .     . 

system, distribution  efficiency 

system,  transmission  effi- 
ciency   

unitooth  wave     . 


113 
399 
136 

192 
205 


3 

18 

193 


104 


402 
63 
52 
68 

363 

313 
61 
58 

368 

6 

20 

111 

37 
281 
287 

283 

287 
444 
470 

474 


INDEX. 


523 


Six-phase  system 434 

three-phase  transformation  .  465 
Slip  of  frequency   converter    or 
general  alternating  current 

transformer 221 

of  induction  motor       .     .     .  238 
Slots  of  alternator  armature,  af- 
fecting wave  shape  .     .     .  384 
Space   diagram    of   transmission 

line 192 

Star    connection   of    interlinked 

polyphase  system     .     .     .  453 
current  of   interlinked  poly- 
phase system 455 

potential  of  interlinked  poly- 
phase system 455 

Starting  of  single-phase  induction 

motor 283 

torque  of  induction  motor    .  254 
torque  of  repulsion  motor     .  361 
Stray  field,  see  Cross  flux. 

Subtraction 490,  494 

Suppression  of  higher  harmonics 

by  self-induction       .     .     .  402 
Susceptance,  definition  .     .     .    '.     fii 
effective,  definition  ....  105 
of   receiver    circuit  with  in- 
ductive line 88 

shunted,  controlling  receiver 

circuit 96 

see  Reactance. 

Symbolic  method,  Chap.  v.      .     .     33 
method  of  transformer     .     .  204 
representation     of      general 
alternating    waves,    Chap. 

xxiV; 410 

Symbolism  of   double  frequency 

vectors 151 

Symmetrical  n-phase  system  .     .  435 
polyphase     system,      Chap. 

xxvi 435 

polyphase  systems  ....  430 
polyphase  system,  producing 

constant  revolving  M.M.F.  436 
Synchronism,    at    or  near  induc- 
tion motor  .  .  258 


Synchronizing  alternators,  Chap. 

xvm 311 

power  of  alternators  in  par- 
allel operation      .     .     .     .317 
Synchronous  induction  motor      .  291 

motor,  also  see  Alternator. 

motor,  Chap  xix 321 

motor,  action  of  reaction  ma- 
chine   377 

motor,  analytic  investiga- 
tion   338 

motor  and  generator  in  single 
unit  transmission  .  .  .  324 

motor,  constant  counter 
E.M.F .  .349 

motor,  constant  generator 
and  motor  E.M.F.  .  .  .329 

motor,  constant  generator 
E.M.F.  and  constant 
power 334 

motor,  constant  generator 
E.M.F.  and  maximum  effi- 
ciency   332 

motor,  constant  impressed 
E.M.F.  and  constant  cur- 
rent   326 

motor  driven  by  induction 
generator 272 

motor,  fundamental  equa- 
tions   339 

motor,  graphic  representa- 
tion   321 

motor,  maximum  phase  dis- 
placement   347 

motor,  maximum  output  .     .  342 

motor,  minimum  current  at 
given  power 345 

motor,  operating  without 
field  excitation  .  .  .  .371 

motor,  phase  relation  of  cur- 
rent   325 

motor,  polar  characteristic    .  341 

motor,  running  light     .     .     .  343 

motor,  with  distorted  wave  .  422 

reactance  of  alternator  and 
synchronous  motor  .  .  .  301 


524 


INDEX, 


Tandem    control    of    induction 

motors 274 

control  of  induction  motors, 

calculation 276 

T-connection  of  three-phase,  quar- 
ter-phase transformation    .  465 
connection     of     three-phase 

transformation      ....  464 
Tertiary   circuit  with    condenser, 
in    single-phase    induction 

motor 287 

Tetraphase,  see  Quarter-phase. 
Three-phase,four-wire  system,  dis- 
tribution efficiency   .     .     .  471 
quarter-phase  transformation  465 
six-phase  transformation  .     .  465 
system,  Chap.  xxxi.     .     .     .  478 

system 433 

system,  balance-factor      442,  446 
system,      distribution       effi- 
ciency      470 

system,  equal  load  on  phases, 

topographic  method      .     .     46 
system,  interlinked  ....     44 
system,  symmetry    ....  436 
system,      transmission     effi- 
ciency     474 

unitooth  wave 389 

Three-wire,  quarter-phase  system  483 
single-phase  system,  distribu- 
tion efficiency 470 

Time  constant  of  circuit      ...       3 
Topographic      construction       of 
transmission    line   charac- 
teristic     -  176 

method,  Chap,  vi 43 

Torque,  as  double  frequency  vec- 
tor       156 

of  distorted  wave    ....  413 
of  induction  motor       .     .     .  246 
of  repulsion  motor  ....  360 
Transformation     of       polyphase 

systems,  Chap.  xxix.    .    .  460 
ratio  of  transformer     .     .     .  207 
Transformer,  alternating  current, 

Chap,  xiv 193 


Transformer,    equivalent    to    di- 
vided circuit 209 

fundamental  equations     208,  225 
General   alternating  current, 
or     frequency     converter, 

Chap,  xv 219 

oscillating  current   ....  510 

polar  diagram 196 

symbolic  method     ....  204 

vector  diagram 28 

Transmission    efficiency   of    sys- 
tems, Chap.  xxx.      .     .     .  468 
lines,  as  distributed  capacity 

and  inductance     ....  158 
line,  complete  space  diagram  192 
line,  fundamental  equations  .  169 
line,  natural  period  of       .     .  181 
lines,     resistance      and     re- 
actance   of    (Phase    Con- 
trol), Chap,  ix 83 

line,     resonance     rise    with 

higher  harmonics      .     .     .  402 
lines  with  resistance,  induc- 
tance,       capacity,      topo- 
graphic characteristic    .     .     49 
Trigonometric  method    ....     34 
method,  limits  of     ....     34 
Triphase,  see  Three-phase. 
Triple    harmonic,    distortion    of 

wave  by 398 

Two-phase,  see  Quarter-phase. 

Unbalanced  polyphase  system  .  431 

quarter-phase  system   .     .     .  485 
three-phase  system  .     .     .     .481 
Unequal      current      distribution, 
eddy  currents  in   conduc- 
tor       139 

Uniphase,  see  Single-phase. 

Unit,  imaginary 494 

Unitooth  alternator  waves  .     .     .  388 
alternator  waves,  decrease  of 

hysteresis  loss      ....  408 
alternator  waves,  increase  of 

power 405 

Unsymmetrical  polyphase  system  430 


INDEX. 


525 


Vector,  as  representation  of  alter- 
nating wave 21 

of  double  frequency,  in  sym- 
bolic representation      .     .151 

Volt,  definition 16 

Wattless  power 151 

power  of  distorted  wave  .     .  413 

Wave  length  of  transmission  line  170 
shape    distortion      and     its 

causes,  Chap.  xxn.  .     .     .383 
shape   distortion   by  hyster- 
esis    .                   ....  109 


Wire,  iron,  eddy  currents    .     .     .  133 

Y-connection  of  three-phase  sys- 
tem   453 

current  of  three-phase  sys- 
tem   455 

delta  connection  of  three- 
phase  transformation  .  .  463 

potential  of  three-phase  sys- 
tem   455 

Zero  impedance,  circuits  of    .    .  506 


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